To begin, stand up straight with your arms extended by your ears. Then, sit back, like you're going to lean back in a chair. As you do this, swing your arms down and back behind you. Your knees should be directly over your feet, not positioned in front of your feet, or you won't generate enough momentum to land on your hands. Swing your arms forward until they reach over your head. Follow your hands with your eyes. At the same time, push through your toes to gain momentum and launch your body backward. Keep your legs together and your core muscles tight. Arching your back too much is called undercutting and will not only make your back handspring look less pretty, but it could also injure your back. As you fall, let your legs continue to drive you upward and backward. Point your toes and extend through your ankles. Make sure you keep your head back between your arms as they approach the floor. Reach for the floor as you fall backwards and make sure to keep your body arched. Keep your arms straight so you don't hit your head on the floor. Your fingers should be pointing up and away from your face with your palms flat on the ground. Let the momentum of your legs, with your lower body still following, carry you over. When your hands are planted on the floor, use the pads of your hands and fingers, your arms, and your shoulders for support. Don't put all of your weight and pressure on your hands. Snap your legs over your head and around to the floor, with  your feet together and your toes pointed. Don't lock your knees, but keep your legs almost straight. Keep your shoulders in line with your hands. Continue by swinging your legs and feet over and snapping them down firmly toward the floor. Keep your upper body straight as your toes approach the floor. To finish the back handspring, land with your legs slightly bent and pop up to release your momentum. Bring your arms straight in front of you and then raise them up over your head. Don't be discouraged if you don't get it right immediately. Work with a spotter until you feel comfortable enough to do it on your own. With enough hard work, you'll have a neat and smoothly-executed back handspring in no time.

Summary:
Bend your hips forward and swing your arms down. Push through your toes and lift your arms up. Continue to fall backwards without arching your back too much. Plant your hands on the floor. Swing your legs over your head. Plant your feet on the floor. Pop up.