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Animation for games falls under two major headings:
Non-interactive elements, such as trailers or the introduction to the game, are fairly straightforward. The animation is done using 2D or 3D techniques and approached in the same way you would create animation for a TV show or movie. The animator usually works from a script and/or storyboards and animates in a linear fashion, from the beginning to the end of a scene or scenes.
In-game animation is a bit different because it isn't a pre-rendered movie, like a trailer. It's animation that happens in real time.
Cutscenes used to be rendered out as movies; however, most games now run them in real time directly from the engine, thus avoiding huge rendered movie files.