Missions are tasks that a nation gives to its chosen adventurers. These extended activities are usually challanging, dangerous, and takes a fair amount of time. In the beginning they can be done alone, but it will soon take an experienced party to complete even a simple-sounding mission.
There are various rewards for completeing these grueling tasks. First, they advance your rank within a city. This makes more powerful conquest items avaialable (talk to elite town guards). High ranks also allow you to advance in the underlying story of the game and rap more benefits. Beyond that, the city often rewards victorious players with gil and other items after specific missions.
Missions are different from Quests. While quests are offered from people (NPCs) in various towns, missions are offered by your country. You do not gain fame doing missions - fame and missions have nothing to do with each other. Instead, missions raise your "Rank" within your country. The higher your rank, the more involved you become in the tale of Vana'diel
Conquest Points are closely related to rank. Conquest Points are gained by slaying monsters while under the effect of the spell Signet. Various NPCs can cast Signet on you - there is always one such NPC by each city gate.
When you reach 1000 conquest points, you will be able to buy your first "Conquest Item." Conquest Items can be bought from the same NPCs that cast Signet on you. Conquest Items are bought with conquest points (they do not cost gil), and usually involve a good solid selection of high quality armor and weapons. Each grouping of Conquest Items has an associated rank - at rank 1, you can buy rank 1 items, at rank 2, you can buy rank 1 and rank 2 items, etc. Note that some of the conquests items are only available to the top nation (the nation whose members have earned the most conquest points) - this is why you don't want to forget to have signet cast on you before you leave the city to fight. You are not merely earning points for yourself to use towards conquest armor and weapons, but you are also contributing to your nation's signet campaign every time you defeat a monster.
As opposed to quests, your missions need to be completed in a specific order in most cases (i.e. you finish mission one, then mission two will be offered). This being said, there are times when you will need to wait to finish a mission (for example, even if you accept mission 6 "Emissary" at level 15, you need to wait until you are at least level 20 to complete it), or times you will decide to wait simply because the missions takes a lot of time and/or traveling to finish. Also, unlike quests, missions are very much story-driven, detailing the return of Beastsmen, the Shadow Lord and the Ancients. Each nation offers unique missions based on their own history and political viewpoints. Each mission begins and ends with a cut scene - as the missions get higher level, some of these cut scenes even begin to resemble short films.