You don't usually 'program animations'.

Yes there ARE different types of animation, some more suited for certain types of animation and implementation systems.

Framed animation: Easy to achieve but poor quality, basically every frame of the object being animated is stored in memory (usually loaded from a file), often a frame might consist of a position, rotation, colour, etc.

KeyFramed animation: Fairly easy, much better quality, this is probaly what you'll use. Instead of storing each and every frame just store the importatn ones and interpolate between them.

Skeletal animation: (Can be combined with keyFramed animation) An object to be animated has an asociated 'skeleton'. Instead of storing a-lot of data about keyframes of the object just store frame data on the skeletons bones as matrices (or whatever, quaternions, vectors etc). Uses much less memory but is more difficult to implement. Basically moving the bones of the skeleton causes a change in the object (ie if it were a person then moving the leg bones would move the leg of the character).

Dynamic animation: This applies to just about any type of animatin that is not pre-determined or scripted. Example: When combined with skeletal animation a characters head could be made to always turn to face the player in a realistic way.

There are lots of different kinds, but these are the ones you'll come across most.

Hope this helps :)