The Riddler is completely obsessed with riddles, puzzles, words games, mind tricks, illusions, etc. He commits crimes but gets an urge to leave clues, his thinking being that his plans are so fool proof that they cannot be stopped, even with clues. However, this is often his downfall, because when Batman and the police are forced to play his games, The Riddler always loses, because no one can outsmart the World's Greatest Detective.
Edward Nigma was a child genius. His father, unable to bear his son's intelligence, beat him out of frustration. Taught to believe that he was an insignificant liar, Edward sought to overcome his inferiority complex by proving his intellect. His downward spiral to crime and delinquency began with cheating during a competition in grade school. Thus began Edward Nigma's career as Gotham's puzzlemaster of crime.
Batman: Arkham Asylum
While The Riddler does not actually ever appear on screen in Arkham Asylum, the game features many riddles and hidden collectibles placed by The Riddler throughout the game. As Batman collects these items and solves the riddles, Riddler calls Batman over the radio. Near the beginning of the game, his comments are insulting and condescending, but when Batman has done most of the tasks, Riddler begins to become angry and accuses Batman of cheating. If Batman completes all of the riddles, and finds all of the items, then Batman determines where Riddler is. He then alerts the police, audio of Riddler being arrested is heard, and it is assumed that he is brought into custody. Several audio tapes can be found in the game that contain conversations between Riddler and one of the psychologists working at Arkham Asylum.
Batman: Arkham City
Riddler serves a similar purpose in Arkham City, as he is still the person who creates the riddles and challenges for Batman, and this time there are several hundred more than in the first game. However, this time Riddler does appear in the game, and he has several challenge rooms hidden across the game, each of which with a hostage (or hostages) for Batman to rescue. If the player completes 400 challenges (either all of Batman's, or a total of 400 between Batman and Catwoman), then Batman learns Riddler's location, and can go to defeat him. As in the first game, he is in contact with Batman over the radio. Like in the previous game, the player can unlock tapes of conversations featuring Riddler, though in this game they are about Riddler hacking into Hugo Strange's communication system and pestering him about Batman and other topics.