A keyboard is made up of keys, as is a typewriter or piano. Most other things are that you press, like to activate something, or perhaps on your computer mouse, are referred to as buttons.

"Key" has a few other meanings. It is the thing that is used to unlock a lock, or to start a car or truck (I think most cars in Europe have keyed ignitions). It can also be the main point of something; "the key the problem is that we to do such and such". "Button" also means the little round things on clothing that go through the little slit (1 cm or so) to fasten things. Not to be confused with "snaps". As for your initial question, Brian's answer is correct.