A market maker is a trader (at a brokerage or bank) who stands ready to buy or sell a given security or derivatives product, thus maintaining a firm bid and offer price. Over-the-counter dealers of this type are called "market makers," while in securities exchange trading they are generally called "specialists." Futures and options exchanges often have market makers assigned to various product listings to assure that a deeper pool of liquidity is available as market participants enter and exit the market. 
The market maker displays bid and offer prices for specific numbers of specific securities, and if the prices are met, they immediately buy or sell from their own accounts. Market makers help to maintain liquidity and efficiency for the securities in which they make markets.
Most firms have a strict separation of the market making and the brokerage sides. Otherwise, brokers would have an incentive to recommend securities in which the firm makes a market.
A Designated Primary Market Maker is a specialized type of market maker, who has agreed to guarantee that he or she will take the position in a particular assigned security, option or option index.