This refers to a democratic “political machine” in New York from the late 1700s all the way through the 1960s. It began as something like a fraternal order, and took the name “Tammany” from an Indian chief. The members started to exert political influence shortly thereafter, picking candidates and influencing elections.
Corruption invariably followed, with various members and leaders taking brides to support causes and candidates they deemed favorable. FDR was very anti-Tammany, and stripped the organization of “federal patronage,” which means that no Tammany members or supporters could be “awarded” positions in the federal government. This reduced the influence of the organization, and it eventually disappeared in the 1960s.