Interboro Rapid Transit (IRT), which built a lot of elevated lines from the Bronx into Manhattan (all the Manhattan lines have been torn down, including the second and third avenue el, which were never replaced with the promised Second Avenue Subway)... they ran shorter, narrower train cars and their remaining lines turned into the 1234567 and Times Square Shuttle (internally, the 0)
Brooklyn-Manhattan transit (BMT), which ran a bunch of trains into Brooklyn...
and the Independent Subway System (IND), which was run by the city with BMT car standards and was responsible for the Manhattan portions of the ABCDEF trains, the G, and the Rockaway Shuttle (internally the H) and bought a few more rail lines in Brooklyn as well (the tail ends of the A and the F).
If you look up the maps, the places where the subway really looks awkwardly constructed, the stations are sprawling, the tracks are at funny angles... it's because they were competitors. Why does the 2/3 cross the L out in Brooklyn without a transfer when the stations are right on top of each other? They were run by competitors. Why do the F/A/C/G shun the other lines as they plow through downtown Brooklyn? Why do you need to walk from the 6 to the F at 63rd Street? Why can't you transfer between the Hewes Street and Broadway (G) stations if they're right on top of each other? Why can't anyone figure out how to navigate Court Street? Why, for years, was there a transfer between the BDFM and downtown 6 at Bleeker St, but not the uptown 6? (That silly little bubble in the subway map.)
To this day there is exactly one spot where an IRT train shares a platform with a BMT train (Queensboro Plaza). And there are essentially two places where trains transfer between the old IND and BMT tracks (at the Manhattan Bridge, and Queens Plaza -- the third, under Central Park between the Q and the F, occasionally figures in maintenance-related detours).
The good news is that New York is lousy with subway stations, and if there's a disruption on one line, there's a very good chance you can still get where you're going on another. The bad news is that's fundamentally confusing.
London's tube stations have a lot of space between their platforms, enough to provide a network of hallways with separate entrance and exit passageways and escalators (some of them are equipped with switchable wayfinding signs so you can have different ways out at morning and evening rush hour). New York doesn't have anything like that anywhere; you'll often have a single set of stairs from the street down to one side of the tracks (with no provision for transferring to the other side at all, to say nothing of elevators). If it's a nice, modern station the tracks will be a little deeper so the station entrance serves as an overpass over the tracks, but other times the transfers will only be available by walking along another subway platform.
http://www.amny.com/transit/nyc-subway-stations-garbage-dump...