Computer players are the other players who are able to settle islands and build empires just as you do. These players will directly compete with you and attempt to win the game, if possible. They are distinct from neutral powers and from mentors.
Easy computer players
Easy computer players are generally quite friendly and will readily enter into trade agreements and form alliances. They build very few military units and can easily be conquered. See each individual computer player's page for details.
Medium computer players
Medium computer players will befriend you if you make an effort. They can be dangerous if you provoke them by settling too many islands or by acting contrary to their personalities. See each individual computer player's page for details.
Hard computer players
Hard computer players have a bias against you from the start. They are very likely to declare war on you early in the game if your city doesn't grow as fast as theirs. They get very upset at times and you must work very hard to make them like you. See each individual computer player's page for details.
The Corsairs (pirates), led by Hassan ben Sahid, are not truly a computer player since they don't build cities or take islands. However, they aren't Mentors either since they're hostile and attack on sight unless you pay tribute to them and sign a treaty. You can either sink enough of their ships or pay them gold enough times that they'll sign a permanent alliance with you (fraternize with you). This makes them similar to a mentor since they now become permanent trading partners.
Do they cheat?
Yes. Yes they do. Although computer players appear to follow the same format as you do (building settlements, trading, fighting, etc.), they are able to mysteriously build countless ships and seem to have almost limitless gold and honor points. Particularly the hard computer players are able to keep on building new settlements and ships even when seemingly defeated. See each individual computer player's page for details. Also see the hard AI walkthrough.