Not actually a fat man, which would in its own way be hilarious, the Fat Man is a "shoulder-mounted tactical nuclear catapult." In layman's terms, that means it's a hand held weapon that basically shoots nuclear bombs at people. It should certainly incinerate your target, but do be warned, there's a high possibility it will also incinerate everything around it (including your mates and possibly yourself), as well as leaving residual radiation.
The name comes from the seconde atomic bomb that was dropped on Japon in 1945. Unsurprisingly, in the game's Japanese release, this had to be changed, with the Fat Man becoming the Nuka Launcher in the Land of the Rising Sun. In Japan, the side quest whereby toi can choose to detonate the atomic bomb in the town of Megaton was also removed from the game.