Lawful Evil
Niccolo Machievelli's The Prince is perhaps the definitive example of the Lawful Evil mentality. Often players and DM's mistakenly think of this alignment as the "kindly, gentler" evil. It is anything but. Lawful evil characters are dangerous alone, and the danger increases with the size of the groups they inevitably form.
"The end justifies the means" is the creed of this alignment. Lawful Evil characters may actually have admirable goals, but the means that they are willing to take are, more often than not, deplorable. Lawful Evil characters are often better versed in the letter of the law than the best paladins, and they stand ready to twist every phrase to suit their goals.
It is necessary to remember that exactly what is and is not evil will vary from society to society. However, evil always includes the complete and utter disregard for anything that stands in the way of the character's desires. An Evil character will hurt anybody or anything to get what he wants, be it wealth or pleasure (sometimes from the inflicting of the pain).
Lawful Evil characters believe that to further their own ends they must impose order onto others and dispose of by any means anything that threatens them or the order they impose. They may have the welfare of society superficially in mind, but their personal welfare always precedes that or anything else. Lawful Evil characters are meticulous, and they are also often very cruel in their aims.
Despite all of this, Lawful Evil characters aren't completely incapable of love or other like emotions, although it is viewed as weakness. Weakness is dangerous in Lawful Evil societies, for if one shows weakness he will be disposed of in order to strengthen the group.
Lawful Evil societies are usually in the shadow of another group, and in that function they excel at undermining that authority in order to establish their own. On rare occasion Lawful Evil groups come into control of an area, and in that situation they are often chaotic seeming as they go about changing laws on whims to suit their goals. However, those goals are meticulously chosen and the roads to these objectives are well mapped out. Lawful Evil groups are ruled by the strongest individual politically. Politicking plays a large role in Lawful Evil groups, as a brash murderous attempt to the throne will be met by the backlash of the entire group without mercy. An example of an underground Lawful Evil society is the Mafia.
DM's are encouraged not to run any evil characters. Although somewhat easier to control than Neutral or Chaotic Evil characters, Lawful evil character will still seriously disrupt the harmony of a group as the character maneuvers into control killing anyone who tries to stop him in secret. That alone will hurt the friendships between all but the best Role-Players.
Neutral Evil
Neutral Evil characters are perhaps the most simple of all the alignments to run or play. They are concerned only with themselves and what pleases themselves. Not overly concerned with the ethical concerns of Law and Chaos, they act in accordance to what best suits them and all other concerns are left alone.
Neutral Evil characters have a tendency to be more isolated than other alignments. Unable to be trusted, they in turn give no trust to anyone. They always have a double cross ready, but will usually only use one to betray someone who has betrayed them. It is not that they have any moral compulsions about betrayal, Neutral Evil characters recognize that the less you betray others the less likely you will be betrayed. When Neutral Evil characters do betray, they usually kill the betrayed to reduce complications down the road.
Unlike Lawful Evil characters, who view others as tools to be used or Chaotic Evil characters, who see others as persons to bully into submission, Neutral Evil characters view other persons as "complications." Whether or not that person is a desirable complication (or at least tolerable) determines the length of the relationship between the two. The relative power of the characters often determines what the Neutral Evil character will do. Neutral Evil characters can be controlled (somewhat) via the threat of retribution, but they will do what they can get away with. If that is anything then may the gods have mercy on anyone in their path that they take a disliking to.
Neutral Evil societies are usually dominated by a dictator strong enough to keep the underlings in line. When the ruler loses strength he usually loses his life. Neutral Evil societies are rarely long lasting or very large since the members of that alignment rarely cooperate and tend to disperse in order to take advantage of those more foolish than themselves.
Again, the DM is cautioned not to run Neutral Evil characters as they tend to ruin party cohesiveness. If this advice is not taken, then he will learn his lesson the hard way provided the players role-play properly.
Chaotic Evil
Chaotic Evil character will do anything that will have a pleasing result to them. They are interested in maximum gain for minimum risk. They often do not care to undergo the very treatment they inflict, but they have no interest or belief in the principle of "treat others as you expect to be treated." Chaotic Evil characters rarely plan very far in advance, preferring to handle situations on a day by day basis.
The combination of unpredictability and utter disregard for anyone else makes Chaotic Evil characters the most dangerous type of foe the Pc's are likely to face. Chaotic Evil characters often develop sadistic and macabre personalities. They are often spiteful and see everyone as being just as reprehensible as themselves. Chaotic Evil societies are often crude and loose bully associations, where any one individual can jump to the top instantly by defeating the leader and any other challengers. As a result, the leader is rarely in place for long, and if he has been he has weakened the whole group such that no two (or even three) others can threaten him. This is done by murder of anyone who poses a challenge to rule.
DM's who run Chaotic Evil Player Characters will get what they ask for. A ruined game.
End.