If you're talking animated cartoons like Daffy Duck, bugs Bunny or Mickey Mouse, then they're created by drawing hundreds of thousands of individual frames and taking a single shot of each of them.
By running the sequence of frames at a specific speed, the appearance of fluid movement is attained.
If on the other hand, you're talking about CGI-style animations (Monsters Inc, Shrek, Avatar), then you're looking at programming the actors to do certain things, not individual frames of animation. This allows much smoother interaction between the actors, their surroundings and the props.
Hair for instance can be fabricated to look extremely realistic and will automatically move depending upon how the actor moves, the wind that's blowing, whether they're wet or dry.
Fabrics too can be modelled to look super-real, unlike with the conventional animating where everything has to be thought about and painstakingly drawn.
Finally, there's claymation, which is about the slowest of them all as the features have to be physically moved for each and every frame.
There are actually other types of animation, but these are the three main ones.