It was developed as a simpler, less costly replacement for the renowned Thompson 1928 "Tommy Gun". It uses the same .45 ACP cartridge and has a firing rate of 600-700 rpm. It was widely used by American military Sergeants in World War II and adopted by the Empire Bay Police Department for use against the gangster presence in the city.
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 recoil that is much less than the Thompson 1928, making it more accurate and giving it better range. It occupies a slot in the top of the weapons HUD along with other automatic weapons.
They're primarily used by Empire Bay police officers, and will be brought out when you've reached a four star wanted level, which is also one of the only ways to obtain one or replacement ammunition. They are, however, used on rare occasions by some enemy late in the game as well in some Mafia II DLC missions.
You can find more real world information about the Thompson M1A1 on its Wikipedia page.
- It is first seen in Chapter 1, where it is used by one of the paratroopers fighting the Germans.