This is the British colloquial (and also official) term for the London Underground. Not all the lines of the tube today are underground, though. However, the early Underground tracks were all subterranean.
The word "subway" can be generally used to describe an underground rail network. In British usage, this word specifically describes an underground pedestrian road crossing.
The New York City Subway is the rail network that serves the City of New York (the subway does not travel outside the city's borders). Originally, the term subway referred to the underground lines, some of which replaced existing elevated and at-grade railroads: (the BMT Subway, the IRT Subway). But as the latter were eliminated in Manhattan and Brooklyn, it came to be used for the network as a whole.
One takes the Subway (or subway) in New York, while one takes the tube (or Tube) in London.
The word "metro" was originally used to describe the Paris rail network (Metropolitan Railroad). Since then, many other major cities have adopted this nomenclature, notably Washington D.C. which had some degree of French influence in its design.
The network in Chicago is fondly referred to as the El (from "elevated") because all the original tracks were above street level. San Francisco has the BART (Bay Area Rapid Transit), which will probably soon be widely written as "bart", as it gets more popular.
While the word subway literally means "below way", many subway systems today combine both surface and underground tracks, as it is not always feasible to have a strictly subterranean network.