Back handsprings are mainly technique and a decent bit of flexibility. There're pretty much 3 main parts to the move, the initial jump, the bridge-like position and then the actual flip. For the jump, you start of standing with arms straight up. Then you squat to just above parallel and bring your arms down and facing forward in front of you. You then spring up and throw your arms back up, which gives the momentum needed for the handspring. What you wanna imagine doing when jumping, is jumping back, instead of jumping straight up. And you also want to try to generate as much power as you can when snapping into the jump and swing the arms back hard to be able to get into position. At this point, you're not going to flip back, but instead just concentrate on getting the jump right. Arms up + standing straight, arms in front + above parallel squat, explode up and back.

The 2nd portion is the sort of back bridge phase. This part, obviously, needs a certain amount of flexibility, so work on being able to get into the bridge position from a standing position. Start off with doing regular back bridges starting from the lying position flat on your back to get used to the movement. Then, as your flexibility improves, you can move onto getting into the back bridge position from a standing position by bending backwards onto a slightly elevated platform. (note slightly, ie not much more than 1 foot, as the higher the platform is, the harder it is to get your hands bent round into the proper position for the bridge) Then gradually reduce the height of the platform as you progress until you can get into the bridge position without a problem.

The 3rd part is the easiest, really. In the back bridge position, you want to push your shoulders slightly in the direction of your back handspring, and flip your legs over. You can start off by kicking one leg over at a time and gradually work to being able to do both. You should come over with your hands and feet on the ground, then you just stand up straight like the starting position in the 1st part.

Now, this isn't really the most conventional way to learn a back handspring, but it's what my coach taught me as I was particularly weak at the floor apparatus and the back handspring itself.

Now for back somersaults or back tucks, you start off in the same way as back handsprings, with your arms straight up. You then squat till parallel or just above and bring your arms down in front of you. And then you explode up and jump. The difference here is that instead of jumping backwards, you jump straight up. At the peak of your jump, what you want to do is to tuck in your knees to your chest as fast you can. This will give a backward turning motion which will flip you around. Landing will take practice as you'll get better the more you do it, so start off on a soft surface. Also, when jumping, instead of flicking your head backwards, you should keep looking straight forward all the way until you reach the peak of the jump and tuck your knees in, by which point you would start turning.