A somersault is one complete revolution or turn in the air, with your feet going over your head. You are not touching the ground during any part of the revolution. A double somersault would be two revolutions, etc. A handspring is a jump off the ground using your hands instead of your feet. A handspring can be forwards, where you are facing the direction of the handspring, or backwards, where you are not. A roll is a revolution or turn while touching the ground, which can either be your feet going over your head or just your back over your chest. A roll can be less than one revolution or several revolutions.

a somersault is usually a roll. a handspring indicates you're touching down with your hands, while a roll or somersault you're likely going over your shoulder.

I think that a PE teacher would be able to demonstrate the difference if you are a visual learner.

roughly speaking, yes the specific kind of somersault would depend on the activity, like, gymnastics, diving, acrobatics..