The Fat Man is a pre-War heavy weapon that launches mini nukes. The mini nuke projectile is very heavy, and if simply fired straight ahead, it will travel only a short distance before falling to the ground and detonating (causing you to be caught well within the very damaging mini-nuclear blast). For optimal range, the Fat Man should be fired in V.A.T.S. mode at high skill levels (which automatically compensates for the projectile's downward trajectory), or at an upward angle to catapult the mini nuke further, so that it impacts at a safer distance.
The Fat Man has an estimated range of 150+ yards. The blast zone will be irradiated for a short time after detonation, giving out up to 5 rads/second.
The Fat Man's condition can wear down surprisingly fast with frequent use (though its firepower is still devastating at any state of repair). It can fire a total of about 167 mini nukes from full condition before breaking.
White House - in the ruins, near a skeleton. Next to the Fat Man are three mini nukes. The White House can be accessed from a Utility manhole at White House Plaza, south of the building on Pennsylvania Ave.
Sometimes the Fat Man (or experimental MIRV), is set up such that using it in V.A.T.S. will cause the Lone Wanderer to aim directly at the enemy, without aiming upwards to compensate for the distance. This means that using the Fat Man in V.A.T.S. could cause the his/her death.
Giving the Fat Man to followers is not recommended, as they may equip it during close quarter fights if facing high level opponents such as feral ghoul reavers or deathclaws. While they won't fire it, it will often get them killed before they can equip a more appropriate close quarter weapon.