The 1x1x1 cube is a cube that is always in the solved state. It cannot be turned in any way because it is only one little cubie. The cube's only purposes are for fun and collecting, although some people say it is the hardest cube to solve. The unofficial world record is 0.02 seconds when the subject had a lucky Cross, F2L, OLL and PLL skip.
How it's made
The cubie is made from a corner cubie of a regular Rubik's Cube. First you need to cut off the part of the corner that makes the cubie be able to turn with the cube. Next, you need to fill the empty spot with apoxie sculpt and mold the sculpt into the corner of the cubie. After the base is done just apply stickers and you have a 1x1x1 Rubik's cube.
Solving the cube
The 1x1x1 is currently the hardest cube to solve, due to its enormous size and the number of steps that lead to its solving. To get a professional's instruction, watch MeMyselfAndPi's solving video. Only the top professionals in the world can do this on their own, and maybe some unknown people that have a secret stash of Rubik's cubes.