The names IRT, BMT, and IND were the names of the three competing transit agencies prior to city takeover in the 40s. The following is a rough guide.
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. Its route followed today's 4-5-6 line from City Hall to Grand Central, then turned west and followed today's shuttle line, and then north at Times (Longacre) Square following the 1-2-3 lines to 145 Street and Broadway. Service to the Bronx was established in 1905 (actually the Bronx portion opened in 1904 from 149 St 3rd Avenue to Bronx Pk as a branch of the 3rd Avenue El, while the Harlem River Tunnel was being completed). The line was quickly extended to Atlantic Ave, Brooklyn, in 1908. The IRT also leased the Manhattan Railway elevated lines - the 2nd, 3rd, 6th, and 9th Avenue Elevated lines in Manhattan and the Bronx for 999 years!
The Brooklyn-Manhattan Transit (BMT, formerly the Brooklyn Rapid Transit, BRT) was the rapid transit company which built, bought, or assumed control of the Brooklyn elevated lines (the Culver, West End, Lexington, Myrtle, Broadway, Fulton St and Fifth Ave Els, of which the Culver, West End, Broadway and part of the Myrtle still exist, and the ground-level extensions to southern Brooklyn, of which the Sea Beach and Brighton Beach were rebuilt for subway service. A portion of the Fulton Street El also remains as it was rebuilt as an extension of IND A train service to the Rockaways in the 1950s.
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 (the Flushing and the Astoria lines in Queens) had trains operated by both companies. The Dual Contracts IRT lines were the Seventh Ave (south from Times Square) and Lexington Ave (north from Grand Central) lines, the Jerome, White Plains Road and Pelham Bay Park branches in the Bronx, and the Brooklyn lines beyond Atlantic Ave. The BMT lines were the Broadway Subway and Nassau Street Subway in Manhattan, the 14th St-Eastern District line from Manhattan to Brooklyn, and Fourth Ave, West End, and Culver lines in Brooklyn. Connections were also made to the company's Sea Beach and Brighton Beach lines.
The Independent Subway (IND) was formed by the City in the 1920s as an "independent" system that was not connected to the IRT or BMT lines. When no private operator could be found, the City's Board of Transportation began operation itself. This system consisted of entirely subway construction with only one elevated portion, a short section over the Gowanus Canal in Brooklyn. The IND lines were the 8th Avenue and 6th Avenue trunk lines in Manhattan, the Queens Boulevard subway in Queens, the Concourse subway in the Bronx, the Fulton Street subway in Brooklyn, the Brooklyn/Queens Crosstown, and the line in Brooklyn via Smith/9th Sts. to Church Avenue. Certain IND lines underpinned existing IRT and BMT elevated lines (6th Av and Fulton St).
The trains of the BMT and IND lines are longer and wider than those of the IRT lines. Therefore an BMT/IND style train cannot fit into an IRT tunnel (the numbered lines and the 42nd Street Shuttle). An IRT train CAN fit into a BMT/IND tunnel but since it is narrower the distance from car to platform is unsafe. Cars from the IRT division are moved using BMT/IND tracks to Coney Island Overhaul Shops for major maintenance on a regular basis.
After city takeover of the bankrupt BMT and IRT companies on June 1, 1940, many of the elevated lines were closed, and a slow "unification" took place, marked notably by establishment of several free transfer points between divisions in 1948 and a few points of through running between IND and BMT lines beginning in 1954 (the connection of the BMT Culver line to the IND at Ditmas Avenue, and the BMT Broadway 60th Street tunnel to the IND Queens Boulevard line). In 1956, the IND connected with the ex-BMT Fulton St El for access to the Rockaways. The Chrystie Street connection in Manhattan, which opened in November, 1967, unified the Manhattan Bridge lines of the BMT with the Sixth Avenue lines of the IND, such that trains from Brooklyn now had access to all of the BMT and IND trunk lines in Manhattan (6th, 8th, Broadway, and Nassau St.). The 63rd Street Tunnel connection will form another link between the Broadway BMT Subway and Sixth Avenue IND Subway and the Queens Boulevard IND Subway (work to be completed by 2001).
Officially, the names IRT, BMT, and IND are no longer used, and the old systems are now designated as the "A Division" (ex-IRT lines), "B-1 Division" (BMT lines), and "B-2 Division" (IND lines), following the Chrystie St Connection opening in 1967. The distinction between the B divisions is blurred because of the unification projects noted above. There is one exception: the IND lacks a number of the sharp curves that one would find on the BMT (such as Crescent St - J, Graham Av - L).