How do I make animations in an iPad app?

Search on the term "Core Animation iOS" on your favorite search engine. You'll find information from Apple's Developer Central site, particularly the https://developer.apple.com/library/ios/#documentation/Cocoa/Conceptual/CoreAnimation_guide/Introduction/Introduction.html#//apple_ref/doc/uid/TP40004514 and https://developer.apple.com/library/ios/#documentation/GraphicsImaging/Conceptual/CoreAnimation_Cookbook/Introduction/Introduction.html#//apple_ref/doc/uid/TP40005406.

There are actually 2 main methods. One is to use Core Animation, if it contains your desired path and animation. The other is to use an animation game loop, where the app periodically calls a routine to redraw the view every frame time. An NSTimer or CADisplayLink can periodically (say at 24 or 30 or 60 Hz) call a routine doing a setNeedsDisplay, which then causes the view's drawRect to be called, etc. The some other periodic code can change some state (moving some X Y button positions, etc.) during or between each frame to provide the appearance of movement or other animation effect, when the view is redrawn. Or OpenGL can be used to redraw some animated 3D world as things move. You can even have each frame change in response to user input. This is the most flexible way to animate, and allows you to customize animations in ways that are impossible for Core Animation, but it uses more power and can be so CPU intensive as to be a lot slower than Core Animation as well.