An auction market is one in which stock buyers enter competitive bids, and stock sellers enter competitive offers at the same time. If this sounds like buying and selling stock on a stock exchange, you're right. It is worth noting that before stocks are traded on a stock exchange, which is referred to as the secondary market, they are created through an initial public offering (IPO). The secondary market is essentially an auction market and is what people are referring to when they talk about the stock market.
Virtually all stocks are traded on stock exchanges. Some exchanges are physical locations, like the New York Stock Exchange (NYSE), where transactions are carried out on a trading floor. The other type of exchange is virtual, like the National Association of Securities Dealers Automated Quotations (NASDAQ), and it's comprised of a network of computers where trades are made electronically.