The names IRT, BMT, and IND were the names of the three competing transit agencies prior to city takeover in the 40s.
The Interborough Rapid Transit (IRT) subway opened in 1904. The city contracted construction of the line to the IRT Company, ownership was always held by the city. The IRT built, equipped, and operated the line under a lease from the city.
The Brooklyn-Manhattan Transit (BMT) was the rapid transit company which built, bought, or assumed control of the Brooklyn elevated lines
Beginning in 1913, the city embarked on a project called the Dual Contracts, under which the city built additional lines that were operated as part of the IRT and BMT systems. Finished mostly by 1920, some of the new lines had trains operated by both companies.