The three major U.S. stock exchanges are the New York Stock Exchange (NYSE), the NASDAQ and the American Stock Exchange (AMEX). As of 2014, the NYSE is the largest and most prestigious of the three. The NASDAQ is a virtual stock exchange.
The AMEX was originally started as an alternative to the NYSE, but as that is now the NASDAQ's function, the AMEX mostly deals with small-cap stocks and their derivatives. Stock exchanges do not own shares but rather serve as a marketplace where stock buyers and sellers can connect. The NYSE is primarily an auction-based format exchange; specialists are physically present on the trading floor.