A supercube is a version of a puzzle where every piece must be returned to its proper position and orientation, even pieces that do not normally have this restriction. For example, a supercube 3x3 would be a puzzle with its centers marked in such a way that the centers must be returned to their proper orientation in order for the cube to be considered solved.
There are very many ways to sticker an NxNxN cube so that it becomes a supercube. The way that was most common in the early days of cubing was simply to print a picture on each side. If the picture is simple enough, rotating the center (on a 3x3) or swapping two or more centers (on a larger cube) will make the picture appear unsolved, even though all of the stickers have been returned to the face they started on. A variation of this is to put some kind of design or logo on each sticker of a face, so that the design is not the same when rotated. This would also force every piece to be in the correct place and orientation in order for the side to look solved (that is, in order for every design on a side to face the same direction). The computer simulator Gabbasoft uses this method to make supercubes: in this case the design is just an arrow.
One other way of creating a supercube is the Pochmann stickering scheme, described on Pochmann's website. The design that he used can be used for any NxNxN cube, as well as similar puzzles such as the Megaminx. Stickers in this design were sold by Cubesmith for a while, but unfortunately they have been out of stock for years. Recently, however, Eastsheen supercubes with a very similar color scheme have been on sale. Buying one of those is the best way to acquire a supercube nowadays.
Solving a Supercube
Solving supercubes is actually similar to solving normal cubes, except that care must be taken to place the correct centers together. It is interesting to note that, unlike on a normal cube, you cannot get away with solving only five of the centers on a supercube - the sixth center must be solved as well. Every method for bigcubes can be slightly modified to solve supercubes, although it will invariably take longer and require more moves than on a normal cube.
There are two main ways of solving a 3x3 supercube. Either you can orient most of the centers as you build the cube (such as fixing all centers but U when building the cross in Fridrich), or you can solve the cube normally and then orient all of the centers later. There are two basic types of algorithms for this. First, you can turn one center by a certain amount, and another center by the opposite amount, using a commutator like (M' U M) E (M' U' M) E'. (This commutator will twist the F center clockwise, and the L center counterclockwise.) Second, you can turn the U center by 180 degrees using either (L R U2 L' R' U)2 or (R' U' R U')5. Note that it is actually impossible to turn one center by only 90 degrees.