Exercises for stretching and strengthening the back

The most common back stretch is the bridge/backbend. You can either do it from standing, where you arch your back until your hands hit the floor, or you can start by laying on your back and pushing up on your hands. You can gradually improve your bridge by pushing your hips up higher, straightening your arms, and walking your hands closer to your feet. Straightening your legs will also stretch out your legs more:

If you have your right/left splits down, there are lots of stretches to help with your arabesque/scorpion/penche/etc. Make sure your hips and shoulders are square and facing the front and arch your back so your head is pointing towards your back knee. Try looking at the ceiling or the back wall. You can also try to grab your back leg (if you can balance) with one or both hands, kind of like a scorpion on the ground.
Now do all of that in the oversplits if you can^ just place a box (yoga blocks work wonderfully) underneath your front foot or have someone lift your front leg up for oversplits.

Another back exercise is basically the inverted version of the bridge. You lay on your belly and push up with your hands (like Ariel the mermaid.) Try keeping your legs together and arching your back until you can see the ceiling or back wall. Or try touching your toes to your head. 

To strengthen your back,
you can do the mermaid stretch^, but take your hands off the floor and try to maintain your arched position while keeping your arms out to the side (second position) or holding them above your head (fifth position.) 
Having somebody hold your feet down or sit on your feet will help you in the beginning. Gradually progress to being able to do it on your own though!
If you can hold your torso upright easily, try lifting each leg at a time and holding it at a 30-45 degree angle from the floor and your other leg. 
Once your back is pretty strong, hold onto a barre or something at chest level and lift your back leg up into arabesque or attitude. You can hold it there OR take a deep plie in your standing leg and pulse your back leg higher each time. Maintain a steady beat and do NOT bounce! Instead try to lift your leg higher each time and not let it just bounce back down!
Try do work on both sides of your back equally. Uneven backs suck because then one leg is so much stronger and more flexible than the other

From: http://ballettips.tumblr.com/