The "Fat Man" bomb used at Nagasaki was not the first nuclear bomb with that name created by Gen. Leslie Groves, J. Robert Oppenheimer, Richard Feyman and pals during the Manhattan Project. That first bomb was tested at Alamogordo, N.M. on July 16, 1945.
Fat Man was an "implosion assembly" plutonium based nuclear bomb. This was the name of both the bomb type program and specifically of F31, the bomb that was actually used. The Fat Man program was considered to be more successful than the Thin Man/Little Boy project (q.v.). This nuclear fission weapon was created with plutonium, unlike little boy which used uranium enriched with U-235.
A Fat Man bomb was not used at Hiroshima because none was ready in time for the mission, thus Little Boy became the first nuclear bomb used against an enemy target.
Fat Man, the bomb F31, was dropped from the B-29 bomber Bock's Car on August 9. The primary target, Kokura arsenal, was obscued by fog and smoke, so Bock's Car turned to the secondary target, the city of Nagasaki. The one gap in the clouds at the at the perimiter of Nagasaki turned out to be the Mitsubishi Arms Manufacturing Plant. It and a large portion of the city were destroyed, resulting in 70,000 direct and indirect casualties. Nonetheless damage was less than if the city center had been bombed as planned.
A second Fat Man weapon, F32, was readied for use in case Japan did not capitulate. The primary target reportedly would have been downtown Tokyo.
Note: The bombing was just 3 days after Hiroshima, the info in the Nagasaki node is erroneous. Some conspiracy theorists claim that Japan was about to surrender anyway, and the Nagasaki bomb was dropped primarily to test the plutonium bomb type.
These relatively tame fission bombs have been largely supplanted by more powerful hydrogen fusion bomb. Isn't that comforting?