The name MP designates Maschinenpistole, literally "Machine Pistol". It's a sub-machine gun developed in Nazi Germany and used extensively by paratroopers, tank crews, platoon and squad leaders, and other troops during World War II.
The weapon has a magazine capacity of 32 rounds and the player can carry an additional 3 magazines for a total of 128 rounds. It has a fully automatic rate of fire with a minimal amount of recoil, making it highly accurate and giving it excellent range for a fully automatic weapon. It occupies a slot in the top of the weapons HUD along with other fully automatic weapons.
In all three Mafia II DLCs they're available for purchase at Harry's shop from the start.
You can find more real world information about the MP40 on its Wikipedia page.
- Though not available until Chapter 9, it can also be obtained from Chapter 2 forward by killing the bartender at the Hill of Tara; however, obtaining additional ammunition will require repeating this.