A "market maker" is a firm that stands ready to buy and sell a particular stock on a regular and continuous basis at a publicly quoted price. You'll most often hear about market makers in the context of the Nasdaq or other "over the counter" (OTC) markets. Market makers that stand ready to buy and sell stocks listed on an exchange, such as the New York Stock Exchange, are called "third market makers." Many OTC stocks have more than one market-maker.
Market-makers generally must be ready to buy and sell at least 100 shares of a stock they make a market in. As a result, a large order from an investor may have to be filled by a number of market-makers at potentially different prices.
To learn more about the basics of trade execution, you should read Trade Execution: What Every Investor Should Know.