A market maker is a dealer who receives privileges in share dealing in return for providing liquidity. This generally means that market makers are required to always quote firm bid and offer prices for at least a minimum quantity of each security they are a market maker in (the minimum varies from security to security).
The London Stock Exchange's SEAQ system requires that all deals are done with a market maker. In return for providing liquidity by maintaining constant quotes, market maker's profit from their bid-offer spread.
London, and some other markets also use hybrid systems which mix market makers with an order driven system in order to guarantee liquidity while reducing spreads.
The responsibilities and the privileges of a market maker vary. The key principle of special privileges in return for maintaining liquidity and an orderly market remains the same. In some US markets the term specialist is used, but this tends to be where the market operates in quite a different way from the British model of market making.