How do a characters designed for a game differ from characters made for books or movies? When a book or a movie is converted into a game, what are the differences and similarities that have to be addressed? Are there pitfalls in character design when jumping between these different medias?
Characters in videogames are strongly defined by what they do, particularly player characters. Lara Croft jumps, grabs, swims, and shoots. Mario's original name is "Jumpman." Megaman bosses are defined by their attacks. In other media, other qualities may come to the fore, such as character history, personality, perspective on life, or inner conflict. For visual media (e.g. animation and film), appearance is paramount, and in audio storytelling formats (e.g. radio drama), just the voice can be a defining characteristic.
Games are about verbs, so when you adapt a character from other media into games, think about what kinds of actions are unique to that character and make him or her distinctive. Spider-Man slings webs. James Bond shoots, drives, quips, and seduces. Transformers... well, you know. One pitfall would be to faithfully reproduce a character's appearance, environment, voice, and even personality to a game, while giving him or her an action set that is irrelevant or even antithetical to his or her behavior from the source material. This may not necessarily result in a bad game per se, but it would probably be a poor adaptation.
This is a huge generalization, of course. Certain genres of film, e.g. Asian kung-fu cinema, are hugely based on verbs. Conversely, non-action qualities are still valuable for game character design. Still, "what does this character do?" is a very useful question to be asking in pre-production.
The chief difference between movies & books and games is interactivity. The player of a game has an active involvement in the world and is (usually) not just an observer of events. Other than this a game character is much like a movie character and should be created and thought about in much the same way. For instance, expressiveness like Sean James mentions. There's also creating realistic dialog, good acting (vocal, if present, as well as physical movements), etc.
With that being said, the main problem in such an interactive environment is keeping suspension of disbelief. It's extremely easy to be knocked out of the game world when a character repeats themselves countless times, walks into walls, generally acts stupid. Care should be given to ensuring that the character behaves properly when interacting with the game's world, environment, and characters.
Also, a lot of care should be taken into considering how the player will interact with this character and how the character should react when the player does whatever they do. If you foresee something the player might try to do to the character and the character reacts in a way consistent with their personality then you have a believable character and the player will be that much more sucked into the world you're creating. For instance, if a player whacks an NPC have them yell "Stop it!" and then "I'm warning you!" and then start fighting back.
Of course, if the game is presented in scripted events (like RPGs), it's even easier.
One of the major differences is that it is much more critical for a player to develop the attitude towards your character that you want them to. For example, if you are developing a character that will be accompanying the player for an entire level, or the entirety of the game, then it is important that the player like the character. If your character manages to get the player to dislike them, then the game will note be much fun.
This was discussed in the developer commentary for Half Life 2 Episode One. There is a character that accompanies the player for the entirety of the game, and it took several revisions before the player actually learned to like that character. In previous revisions the player would come to find the other character annoying, which made the game experience not very fun.
A second note is that characters in videogames (or movies) must be much more expressive. While in a book you often get a chance to go inside a character's head, this is not as much of an option in a game. Your characters must explicitly say what they are thinking or feeling or you risk the player not getting the information you want them to.