The Puzzle Cube (3x4x4), along with its sibling, the Puzzle Cube (3x3x4), were among the first cuboid variations of the original Rubik's Cube to be invented. You can think of the 3x4x4 Rubik's Cube as a Rubik's Revenge Cube (4x4x4) that is missing a layer on one of the axes (usually the Y-axis between the white and yellow faces, as presented in the 3D simulation on this page). The 3x4x4 cuboid puzzle was originally invented by Tony Fisher back in 1998. It was initially hand-made using the mechanism of a Rubik's Revenge Cube (4x4x4), however, since 2014 this puzzle is mass-produced by a company named Ayi Liu, which made 4 different color variations of it (black, white, blue and transparent). Several brands manufacture this puzzle today, but even though this puzzle is so closely related to the Rubik's Revenge Cube, it is not manufactured by the Rubik's brand.
Like the Tower Cube (2x2x3) and other cuboid puzzles, the top and bottom faces of this puzzle cannot be mixed with the side faces. This is because the 3x4 layers of this puzzle can only make double turns (180 degrees) while the 4x4 layers can also make regular 90-degree turns. That unfortunately means that algorithms you might already be familiar with from solving a regular Rubik's Revenge Cube are usually not applicable to this puzzle. This fact, however, makes it easier to solve than a regular Rubik's Revenge (4x4x4) due to the smaller amount of possible moves and combinations.