The topic you are asking about is skinning. There are lots of resources on the web about it. It can be done on the hardware or on software. A software implementation is probably the easiest way to start, then you have something to benchmark any hardware solution against.

It sounds like your particular example could be covered with rigid skinning, where each vertex is influenced by exactly one bone. However, if you're going to the trouble of implementing skinning you're probably best off going straight for smooth skinning, where each vertex is influenced by N bones with a relative weighting for each bone.

Just google "skinning and animation" (googling 'skinning' on its own tends to get you information about how to skin dead animals) and start reading. It's a big topic and it's not a trivial thing to implement.