The Fisher Cube is a shape-mod of a 3×3×3, meaning it has the same mechanism as that cube. It was made by Tony Fisher and is the cube Tony Fisher is most known for. Basically, it is a standard 3×3×3 which has had its cuts rotated 45° about the Z (white-yellow) axis, so that the red-green-orange-blue edges and centres are swapped, and the white-yellow edges and corners are swapped.
Thus the solve is very similar to that of a regular 3×3×3, but with centre orientation visible and edge orientation invisible on the red, green, orange and blue faces.
When you get close to the end, there is a parity that kind of looks like a 4×4×4 parity As noted above, this is because you have flipped one (or three) of the middle layer edges, hence this problem is easily solved using ordinary 3×3×3 algorithms.
When you do a U-perm 3 times while the yellow side or white side is on top, it rotates the center 90 degrees. It also twists the center above, but that really doesn't have a visible effect.
This cube is mass-produced by YJ.A similar cube is called the Windmill Cube. It is also a shape-mod of a 3×3×3, but instead of a 45° angle, it is shifted at a different angle so that some of the cuts pass through the edges of the cube.