It was designed for the US military as a cost effective alternative to other more expensive automatic weapons, optimized for cheap and fast mass production with a simple design to be used in the European war effort.
The weapon has a magazine capacity of 30 rounds and the player can carry an additional 3 magazines for a total of 120 rounds. It has a fully automatic rate of fire with a moderate amount of recoil, making it better when fired in short bursts. It occupies a slot on the top of the weapons HUD along with other automatic weapons.
They first become available in Chapter 8 where they are carried by some of the Greasers at the Empire Bay Forge. They later become available for purchase at Harry's Gun Shop in Chapter 14, meaning replacement ammunition remains scarce throughout most of the game.
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 Grease Gun on its Wikipedia page.
- While technically named the M3 submachine gun, it got its grease gun nickname due to its close resemblance to the common mechanic's tool.