The word usually translated ‘apostle’ in the New Testament is ‘apostolos’ which in Classical and Hellenistic literature means, ‘messenger, envoy, ambassador’, devoid of any religious connotation, from the verb ‘apostellô’ meaning, ‘to send off, dispatch on some mission or service’.

Jesus as One who was sent
Jesus is called 'apostolos' in the New Testament only once , namely, in Heb 3:1, "the messenger (apostolos) and high priest of our confession," and is immediately compared to Moses, another personage appointed and sent by God, but to whom Jesus is superior as 'son' is superior to  'servant' (vv.5-6). In the Gospel of John the sending of Jesus by God is a main theme of the author, yet, the term 'apostolos' is not employed for Jesus. John, in contrast to the Synoptics, where the subject is the sending of Jesus by God, prefers the verb 'pempô' to 'apostellô', -both meaning 'to send'- especially in the description of God as 'he who sent me' ('ho pempsas me', 6 times).

Paul: 
As the Pauline texts are by far the earliest texts to deal with the Christian messenger concept, we must  analyse them first.  
Paul,  in the salutation of his letters,  introduces himself as 'apostolos' i.e. messenger, envoy  of Christ /Christ Jesus ' (1 Cor. 1:1; 2 Cor. 1:1, etc.)  In Rom 1:1 and 1 Cor. 1:1 he stresses the calling, 'called to be an envoy' of Christ  and in Gal 1:1 he emphasises that his envoyship is 'not from men... but through Jesus Christ and God the Father.'  He claims he was set apart before he was born, was called through God's grace and had a revelation of the Son  for the task set for him, to preach him among the Gentiles (Gal 1:15f.). In Rom 1:5 and 11:13  he insists he received his 'envoyship' as grace to be 'envoy to the Gentiles'.

Paul vindicates the parity of his call with that of the other 'apostoloi'. He doesn't claim to have known Jesus in his lifetime, a condition required of the other 'Apostoloi' according to Acts, but he insists that he has seen the risen Christ like them, 1 Cor 9:1f., " Am I  not free? Am I  not an envoy ('apostolos')? Have I not seen Jesus our Lord? Are you not my work in the Lord? 2   If I am not an envoy ('apostolos') to others, at least I am to you; for you are the seal of my envoyship (apostolê) in the Lord. " This vehemence shows that his envoyship was questioned, and he defends it especially with respect to the Corinthians. Thus in Paul's view envoyship is not a universal office but a function in relation to a group for which the commission is given ( cf. Gal 2:8 and 1:16) 
  Again 1 Cor 15:5-9  " and that he (Christ) appeared to Kephas, then to the twelve. 6   Then he appeared to more than five hundred brethren at one time, most of whom are still alive, though some have died. 7   Then he appeared to James, then to all the  envoys ('apostoloi') 8   Last of all, as to one untimely born, he appeared also to me. 9   For I am the least of the envoys ('apostoloi'), unfit to be called an envoy ('apostolos'), because I  persecuted the church of God. " In v.7 'apostoloi' embraces all those who have seen the risen Christ and does not refer to just the 'Twelve' already mentioned in v. 5 . Thus for Paul there are more 'apostoloi' than the 'Twelve'. One may conclude from this passage that Paul considers all those who received an appearance, whether physical or spiritual,  of Jesus are  'messengers, envoys' of Christ to witness to him.
Paul's envoyship  to the Gentiles is as valid as that of Peter to the Jews (Gal. 2:8).  He compares and places himself on the same level as the 'apostoloi' in Jerusalem (Gal 1:17, 19) and in 1 Cor. 9:5  Paul claims the same rights for an accompanying wife as "the other envoys  ('apostoloi')  and the brothers of the Lord and Kephas"; people mentioned as recipients of an appearance of Christ in 1 Cor 15

Paul doesn't seem to be the originator of the term and concept since he refers to others who are 'apostoloi' before him (Gal. 1:17) and some of whom he met or did not meet in Jerusalem, Gal. 1:19   "but I did not see any other envoy except James the Lord's brother." This is understood by some as indicating that Paul, here, seems to consider James as an 'apostolos' /envoy. - especially when considered in conjunction with 1 Cor 15:7.

Thus Paul's definition of 'apostolos' would probably have been, 'a person who has had a revelation of the risen Christ and was commissioned by Christ himself to preach and witness for him.' 

In 2 Cor 12:11 he defends his envoyship again  not so much in relation to the first 'apostoloi'/envoys whom he seems to hold in high esteem but against others, the Judaizing missionaries, called 'apostoloi'/envoys by those questioning his envoyship and whom he could sarcastically call 'super-apostoloi', "I have been a fool! You forced me to it. Indeed you should have been the ones commending me, for I am not at all inferior to these super-envoys, even though I am nothing."  

In Rom 16:7 Paul explicitly refers to two people, Andronicus and Junia, as envoys,  "Greet Andronicus and Junia, my relatives who were in prison with me; they are prominent among the envoys ('apostoloi'), and they were in Christ before I was." These are plausibly a couple, man and woman and perhaps to be included among the many early messianists who claimed to have seen the risen Christ as in 1 Cor 15:6-7.

In 1 Thes 2:6 "we might have made demands as envoys of Christ," the 'we', according to some, may include Silvanus and Timothy mentioned at the beginning of the letter. - but Paul does not explicitly call them 'envoys' and it is seriously questioned whether Paul would  position these people on his own level of envoyship after all the trouble he goes through to prove himself an 'envoy' (apostolos') like the earlier ones unless they were among those who have seen the Christ, which surely does not apply to Timothy, Paul's child in the Lord (1 Cor 4:17). 1 Cor 4:9,  is another 'we' passage wherein Apollos is seen by some commentators to be included, however as in the previous passage the 'we' means chiefly Paul himself as the following listing of tribulations clearly indicate but at the same time he associates himself with the troubles to which other 'envoys' were also submitted.

From Paul we gather that, though the envoys, as the chief heralds, were pivotal for the communities at the beginning, the vitality of the community was demonstrated also and  concomitantly by other functionaries and people gifted with diverse charisms, such as prophets, teachers, healers, leaders etc. 1 Cor 12:28f:   "And God has appointed in the community first envoys, second prophets, third teachers; then deeds of power, then gifts of healing, forms of assistance, forms of leadership, various kinds of tongues...." (cf. also Rom 12:6-8, 1 Cor 12:7-11,  and Eph 4:11).  This diversity of functionaries and charisms is a truer picture of the actual situation in the operation of the Christian communities in the time of Paul than an assumed monarchical and hierarchical order.

The Gospels and Acts

Mark:  In Mark the term 'apostolos' is  rare and questionable. In Mk  3:14   "And he appointed twelve, -whom he also named 'apostoloi'-, to be with him, and to be sent out to proclaim the message..." the expression, "whom he also named 'apostoloi' " is seen by many critics as an addition after Lk 6.13.
When Mark resumes the account of the sending of the disciples he starts, "The messengers ('apostoloi', those that were sent) gathered around Jesus, and told him all that they had done and taught (Mk  6:30 ). The word 'apostoloi' here, since the author does not use it anywhere else as a title would, most likely, not be understood as such by his original readers but they would take it in its common meaning, i.e. messenger, emissary, one sent for a purpose. Thus the Markan community did not make use of 'apostolos' as a title.

Matthew: Matthew, for the disciples of Jesus, besides the term 'disciple/s' prefers to use 'the Twelve'. The term "apostolos" is used only once, at the introduction of the list of the twelve, Mt. 10:2   "These are the names of the twelve ‘apostoloi’: first, Simon, ..." This is an editorial introduction which could easily contain a later interpolation of the word 'apostoloi' [note: Some MSS  have 'disciples']. So Matthew's community definitely did not make real use of this term.

John: In John the term 'apostolos' does not figure at all in connection with this topic. In Jn 13.16 it just means 'messenger' and does not apply to Jesus' disciples.  Mark, Matthew and John , all employ 'the twelve' for the inner circle of  disciples.

Luke-Acts: The situation is different with Luke-Acts. Luke-Acts' concept of 'envoyship' is definitely different from Paul's.

Luke: In the parallel passage to Mark and Matthew above, dealing with the choosing of the Twelve, Luke introduces the title 'apostoloi,'  which he subsequently employs in the Gospel and further develops in Acts. - Lk. 6:13   "... he called his disciples and chose twelve of them, whom he also named 'apostoloi' ‘Envoys’." The word 'named'  indicates that Luke here turns the term into a title. Then again Lk. 9:10   "On their return the envoys told Jesus all they had done."  "apostoloi" is used for the ‘Twelve’ three more times in Luke: Lk. 17:5; 22:14, and 24:10 . It is significant that the term is editorial and not on the lips of Jesus; however in 11:49,(a Q saying parallel to Mt 23:34) Luke substitutes 'apostoloi' for Matthew's "sages and scribes" and refers to the future, which perhaps indicates Luke's predilection for this word. As  the other Gospels do, Luke, in addition to 'apostoloi', also uses 'the twelve' for the inner circle of disciples. 
In the passage narrating the Sending of the Seventy, 10.1ff  (exclusively Lucan), the term 'apostoloi' is not used even though these are sent out for a similar mission as the 'Twelve'. - this corroborates the view that  in Luke's mind the title 'apostoloi' is reserved for a specific group, i.e. the Twelve.

Acts:  For Luke only the Twelve ( or the Eleven) are 'apostoloi'. In his view the early Christian community is represented by them. In Acts 1.2 he reiterates their choice by Jesus  stressing that this choice was through the Holy  Spirit before Jesus' departure and that they shall receive the Holy Spirit to be Jesus' witnesses ( vv. 4 and 7f). This is the essence of 'Envoyship'  in Luke's ecclesiology. He develops this concept in 1.15-26 by reestablishing the 'Twelve' through the divine choice, indicated by lot, of Matthias in place of Judas Iskariot, and by defining the nature of 'envoyship' , that is, a man must fulfil two conditions to qualify as an 'apostolos', a) to have known Jesus during his lifetime  from the baptism of John  to his ascension and b) to be a witness of the resurrection together with the 'Eleven' (1.21-22). This situation is unique, it cannot be reproduced. Thus according to this author's mind 'envoyship' cannot be handed down to others, in other words there is no apostolic succession.   There is no transfer of 'Apostolic' authority in the choice of Matthias. It is worth noting that there is no imposition of the hands here, the appointment is left in the hands of the Lord (God or Jesus). 

This reestablished group is exclusively called the 'apostoloi' not the 'Twelve' in Acts (except once, 6:2).  It is on this group and the other disciples of Jesus that the Spirit descends at Pentecost enabling them to start their ministry as the guarantors of the resurrection. The title of 'apostoloi' is so restricted to this initial group that the author makes no mention of an appointment of a successor to James, the brother of John,  killed by Herod (12.2), thus indicating that the 'Apostoloi'  are not replaceable as they die;  the divine choosing of Matthias was done to complete the 'Twelve' before the witnessing was launched. Also, despite all the effort to present Paul as the great missionary, repeatedly commissioned by Christ to preach,  Acts does not refer to him as the 'apostolos' of the Gentiles. For Luke, Paul's meeting of Jesus on the road to Damascus (despite the triple narration)  does not have the same value as Jesus' appearance to the 'Apostoloi' prior to the ascension; it does not qualify him for the title of 'Apostolos.' The one (or two?) time Paul and Barnabas are referred to as 'apostoloi'  in 14:4, (14, unattested in some MSS)  the term may have been  used in the common meaning of 'messengers' . 

The 'Apostoloi' are presented as the first leaders of the Jerusalem community of believers and as the launchers of the Messianic movement. They proclaim and witness the resurrection (4.33), their teaching is the  foundation of the  believers' faith (2:42), they perform 'wonders and signs' (2:43, 5:12), they receive the offerings of the other members in aid of the community (5:35) and they appoint the Seven (6:6) as table-servers.  A special function Luke attributes to the 'Apostoloi' is the imparting of the Holy Spirit by their prayers and laying-on of hands when baptism alone is ineffective in this respect (8:16, 17). Note that Luke does not attribute to them any priestly functions.

In chapter 15 we find that the 'Apostoloi' are not any more the sole leaders in Jerusalem, they are paired with the 'elders' (15:4, 6, 22, 23; 16:4). This is an  indication that Acts does not consider the 'elders' as the replacement or successors of the 'Apostoloi' but they are coexisting with them and concurrently  performing leadership functions. This chapter also reveals the presence of a special personage, neither listed with the 'Twelve' nor established, anywhere in the book, as a successor to the 'Apostoloi', but very prominent in the governance of the Jerusalem community, namely James (the brother of the Lord) who plays a considerable part in the decision regarding the admittance of Gentiles in the movement (15:13ff).

After 16:4 the "Apostoloi' disappear from the scene in Acts and in 21:8, when Paul visits Jerusalem he deals only with James and the elders.

Although Luke did not invent this concept, his presentation reflects the idealised picture of the 'Apostoloi' in vogue in his own time, the last decade of the 1st century.

The concept of the "apostoloi" ('messengers, envoys, those sent with a task') as a special group probably developed from the belief that Jesus had, somehow, commissioned his disciples to bear witness and to preach as expressed in Mt 28:19f ('make disciples of all nations'), Lk 24:47f ('be preached.. to all nations'), Jn 20:21 ('as the Father has sent me, even so I send you'), Acts 1:8 ('..you shall be my witnesses ...to the end of the earth') and the longer ending of Mark, Mk 16:15 (Go... and preach the Good News to the whole creation.') Despite this order to preach, however,  in the New Testament , there is an eloquent lack of evidence  of any missionary activity of the Twelve outside of Jerusalem and the preaching/witnessing we get in Acts is represented only by Peter  - at times accompanied by the silent John.   This concept of the 'messenger' was combined and compared to that of the Old Testament  prophets who were initiated into their preaching career by a vision. Thus the 'Apostoloi' and Paul had the vision of the risen Christ start them on their preaching mission.  

It is highly improbable that Jesus ever called his disciples 'apostoloi' or sent them on any mission during his lifetime;  the sending of the twelve to perform healings during his lifetime is a later development; not even the healings of Jesus himself are verifiable as historic events let alone those of his disciples. Jesus was the one with a mission and the disciples were the recipients, beneficiaries of his mission,  but the later generations anachronistically retrojected the idea of ‘apostleship’ into the time of Jesus. Perhaps already in the last decades of the 1st century the rabbis had started the custom to send some of their disciples as their representatives ('shaliahim' i.e. 'those sent' i.e. 'apostoloi' ) as evidenced in the 2nd century CE,  to conduct their business, financial or religious, in their place, which may have encouraged the Messianic movement to enhance the 'apostolos' concept on the pattern of the Rabbinic 'shaliahim'.

Acts 14.23 has Paul and Barnabas appointing 'elders' in some communities they established. Far from being a case of  transmission of  characteristics or powers from one person to another, this means helping the local community set up their leaders to administer and care for the community. Surely, each community required order and governance but hardly a resident 'Apostolos - messenger'.  If Paul or others actually appointed elders  (the only time this is indicated is  Acts 14:23)  to lead the community, this in no way means that they hand down the distinctive feature of 'apostolos' to these elders but that they confirm them in the leadership of their local community. If Paul ever did this he certainly could not have passed on his 'envoyship', a special personal gift, which he boasts to have acquired by virtue of his seeing the Lord who called him for a specific task. This ‘envoyship’, this special status as witness to Christ’s resurrection is not transferable or bequeathable.  In all honesty, it should be admitted that the New Testament never shows any of the Twelve or 'Apostles' appointing a successor.

Other Writings
The content of the term 'apostolos'  is theologically varied. 
1)  Hebrews uses it  of Christ (Heb 3:1).

2)  Luke-Acts and Paul see in it the witness to whom the risen Jesus appeared  and who was commissioned to preach. Luke emphasises the eyewitness to the life and resurrection of Jesus while Paul stresses the  appearing of Jesus to him personally.

3)  In later writers the ‘Apostoloi’ appear as a body belonging to past history: in 2 Pet. 3:2 and Jude 17, not necessarily the 'Twelve', they are the representatives of normative preaching.   Ephisians pairs 'apostoloi' with prophets as the foundation and agents of revelation (Eph, 2:20; 3:5). Thus here the foundation is the message of earlier time irrespective of the source, whether imparted by Jesus or the Holy Spirit.  In the Pastorals 'apostolos' is reserved only for Paul as the only guarantee of the tradition (1Tim. 1:1, 2:7 , 2 Tim. 1:1, 2 Tim. 1:11, Titus 1:1)   In Rev. 21:14 the term refers eschatologically to the ‘Twelve’,  " And the wall of the city has twelve foundations, and on them are the twelve names of the twelve 'Apostoloi' of the Lamb. " This is akin to Luke’s having the Twelve perform the eschatological function of sitting with Jesus in judgment of the twelve tribes of Israel (Lk 22:30).

4)  False 'apostoloi' - in the wider sense the word also refers to false 'envoys'. In Paul's time these are already pointed out, thus  2 Cor 11:13   "For such boasters are false envoys, deceitful workers, disguising themselves as envoys of Christ." The later writing Revelations, still makes reference to similar people (Rev 2:2) 

This variety in connotation of the term ‘apostolos’ does not correspond with the meaning of the word ‘apostle’ in the modern languages which conveys an exclusively religious and distinctively Christian sense acquired and accumulated during the process of the development of the Christian Church. The early readers of Paul and of the other New Testament writings did not understand ‘apostolos’ as we understand ‘apostle’ today. Therefor it is about time to realize that the translation ‘apostle’ in the New Testament passages is  anachronisticaly misleading.