My mom, who played piano extensively when she was younger, insists that I shouldn't curve my fingers while I'm playing–my fingers should be straight and the pad of each finger should touch the key. My piano teacher tells me to curve my fingers and hit the keys with the tip of the finger. Who is right?
Your mom is wrong I'm afraid. Theoretically your fingers should be curved to the extent this is physically possible. How well you can do this depends on the span of your hand, as the more you stretch open your hand – the flatter it will become.
If you play with flat hands (and some great pianists have played that way) it will reduce your power and dynamic range and increase the likelyhood of injury.
The tip of your finger should make contact with the key – not the pad.
This is what I was taught, but I guess there are two schools of thought. If you learn to play your way you can still and will from time to time play with your hands flat, however it is much harder to go the other way (from flat to curved).