The MP5k takes the place of the MP5 from Call of Duty 4: Modern Warfare.
The "K" in the name is a shortened version of the German term "Kurz", which is English for "short".
The MP5K has relatively high recoil and muzzle flash, which means that it is quite difficult to use effectively at longer ranges.
In Single Player and Spec Ops, the MP5K has no stock, though in online it does. Oddly, the MP version, with the stock, has more recoil. This is likely to balance the game.