Puzzles like this may be called Skewbs (sphere/cube) At left is a good shot of both sides of the same puzzle ball. Shot by SpeedCubing's collection. 

These are 3D varients of the old "Sliding blocks" puzzle. There are various picture-areas that you can arrange only by sliding the pieces in certain ways. Often frusterating, but still interesting to collect. Cool to note that the images on the balls are 3D also, a step that few take. The green one is un-arranged, but it appears to have Tails as well.

Here's the puzzle ball in it's box! So this is officialy "Sonic's Puzzle Ball". There are questions from Tails on the side, but you can't read them. Who produced these, and when? The ball itself isn't so rare, but one in the box, certainly is!
Be sure to visit the website below, it has a great description of it.
Do you want to buy the puzzle ball?
You might be able to, for about 16-20 dollars. Steven Matarazzo has discovered Mefferts Puzzles, which sells these right now! They CAN go in and out of stock though, so don't count on being able to get one at all times, but if you presist, you should be able to get this for yourself.