You can think of the Floppy Cube as basically the middle layer of a Rubik's Cube. The puzzle consists of 9 cubes ordered as a 3x3 square grid and colored accordingly. It was originally invented in 2004 by Katsuhiko Okamoto, a Japanese inventor specializing in Rubik's Cube modifications. The puzzle is manufactured by Gentosha Toys since 2009.
So, why is it called the Floppy Cube? The name "Floppy Cube" comes from the fact that in order for the mechanism to function properly, the middle square bends slightly upward and downwards to allow for the turning of the side pieces. The wonky shape of the center piece while twisting the edge pieces is what gives the puzzle its name.
The interesting thing about the Floppy Cube is that its edges can flip but not move, while its corners can move but not be flipped in place (their orientation is determined by their position). This fact results in a very small amount of possible combinations for this puzzle - only 192. A quick computer analysis shows that while most of these combinations can be solved in 4 or 5 moves, the maximum amount of moves required to solve this puzzle is 8 (for the single case of the superflip pattern).
In 2009, the inventor of this puzzle, Katsuhiko Okamoto, also invented its successor, the "Super Floppy Cube". The Super Floppy Cube is almost identical to the Floppy Cube only it also allows "jumbling" - turning the edges independantly from each other, making it possible to move the corners around and even put all of them around a single edge piece. This of course makes the puzzle lose its "flat" appearance and you can scarmble it into different shapes. In the year 2019, the Rubik's Brand created their own version of the Super Floppy Cube and named it the "Rubik's Edge". Counterintuitively, the Super Floppy Cube (or Edge Cube) is not harder to solve than the original Floppy Cube. On one hand jumbling allows it to be scrambled in over 3 million different combinations (comparing to only 192 on the original Floppy Cube), however this degree of freedom to rotate the edges independantly from each other also makes it much easier to solve.
Tip for solving the Floppy Cube: start with bringing a single corner piece to its right location, then continue by flipping both adjacent edges to their right orientation, keep making your way around the puzzle from both sides by solving their adjacent corners etc. until you've made your way to the last piece and the puzzle is completely solved.