The Square 1 is an example of a bandaged cube. What this means is that you can't turn it in as many directions as you can a normal cube. The cubies are not all the same size and, once you start turning it, you end up with all sorts of different shapes. It was patented in 1993 and has been around for a while.
The Square 1 is a tricky puzzle. Solutions involve turning faces 30° at a time and can get quite complicated to follow let alone learn. Notations for these solutions usually use numbers. For example, (1,3) / means that you turn the top face through 30° clockwise, the bottom face 90° clockwise and then turn R2. I find this hard to remember.
Without a little guidance, this puzzle can reduce an experienced cuber to tears quite quickly. This puzzle is worth getting when you feel like a bigger challenge than you've had so far. The first challenge is to return the scrambled puzzle to a cube shape and then to place the pieces. It is hard to solve without, at some stage, having to disrupt and reform the cube shape. Square 1 algorithms that use multiples of 3 can often be used on the dihedral puzzles for edge and corner swapping.
These are available to purchase new and second-hand ones are around - not surprisingly, many of the second-hand ones have had little use by their owners.