I know this sounds like a stupid question, but here goes anyway. How do you do animation?

For example, I might click on a button, and I want that button to do a one second animation. Or I might click on an army icon, and then that icon moves from the left end to the right end of the screen. Or I might click the "Execute" button and a bunch of animations take place on the screen.

I'm using OpenGL, but what I'm asking for is generic advice to incorporate animation into my game, independent of the graphics library (e.g. GDI+, SDL, OpenGL, DirectX) I use. Surely there are some library-independent aspects to animation?

Here's my game loop, by the way, pretty standard stuff:

while Game.Running do begin
Game.HandleEvents;
Game.Update;
Game.Render;
end;

I hope that makes sense.