Metro
Metro is a general term for an urban transportation system, which can include underground train systems but is not limited to it. Other components can include wheeled busses and light rail systems that run on streets. Heavy rail runs on dedicated tracks, which may be at ground level or elevated above it, as in the "El" of Chicago. A few systems have more exotic transportation components such as monorails and magnetically levitated trains.
An underground urban component is called "underground" in British English, but may be called "Metro" in many languages, and "subway" in American English.
Metro systems often interconnect with long-haul transportation systems, including bus, train and air.
Paris Metro
New York transit system
Bay Area Rapid Transit
Bay Area Rapid Transit is a heavy rail system for the greater San Francisco Bay area.
Washington, D.C. Metrorail
Operated by the Washington Metropolitan Area Transit Authority, Metrorail is a heavy rail system organized into five color-coded lines. The lines interconnect at various points; some of the stations (e.g., Pentagon Building) are also major Metrobus stations. It serves Reagan Washington National Airport and connects, by express bus, to Dulles International Airport and Baltimore-Washington International Airport. Metrorail interconnects with regional (Virginia Railway Express (VRE), Maryland's MARC) and national Amtrak train services at downtown Union Station and several suburban stations.
Metrorail drew many of its design inspirations from BART.