The Puzzle Cube (3x3x4), along with its sibling, the Puzzle Cube (3x4x4), were among the first cuboid variations of the original Rubik's Cube to be invented. You can think of the 3x3x4 Puzzle Cube as a normal Rubik's Cube (3x3x3) with an additional layer in one of the axes (usually the Y-axis between the white and yellow faces, as presented in the 3D simulation on this page). The 3x3x4 cuboid puzzle was originally invented by Tony Fisher back in 1995. It was initially hand-made using the mechanism of a Rubik's Revenge Cube (4x4x4), and only a small number of them were sold. This puzzle started to be mass-produced in 2009 when a company named Cube4You decided to raise the glove. Several brands manufacture this puzzle today, but even though this puzzle is so closely related to the Rubik's Cube, it is not manufactured by the Rubik's brand.
Like the Domino Cube (2x3x3), 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 3x3 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 Cube are usually not applicable to this puzzle. The good news however are, that this puzzle has significantly less possible combations - around 41 quadrilion. This is still a huge number but it is dwarfed by the Rubik's Cube 43 quintillion possible combinations.
A common approach to solving this puzzle is "from the inside out" - start by solving the edges of the side faces, then the corners around them, and finish by solving the top and bottom layers.