The NYSE and AMEX are both primarily auction based, which means specialists are physically present on the exchanges’ trading floors. Each specialist «specializes» in a particular stock, buying and selling the stock in a verbal auction. These specialists are under competitive threat by electronic-only exchanges that claim to be more efficient (that is, execute faster trades and exhibit smaller bid-ask spreads) by eliminating human intermediaries.
The NYSE is the largest and most prestigious exchange. Its listed companies represent about $18 trillion in market capitalization. AMEX is a smaller but quite prestigious exchange. AMEX also has a history of innovating: it pioneered the concept of exchange traded funds and it has the second largest options trading market.