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An institution or individual that quotes bid and offer prices for specific stocks or other marketable securities that it holds in inventory (often referred to as 'makes a market in'), and is prepared and able to buy or sell those securities at any time on its own account.
Note that this is not necessarily the same as a brokerage house, as a broker typically acts as an agent for a customer and charges a commission for helping to consummate a trade with another investor (rather than acting like a market maker who serves as a 'principal' by buying or selling from its own inventory of securities).
Market makers are normally tasked with providing sufficient liquidity in order to reduce volatility in prices and maintain a 'fair and orderly market' for stocks. For example, they will typically buy a stock when there are few or no other bidders in the market yet many investors wish to sell this security. By doing so, market makers can instill greater investor confidence in the financial markets and encourage these investors to send more orders to these trading centers. This can create a virtuous cycle where these additional orders result in greater depth and liquidity which, in turn, attract even more investors and orders. Thus, market makers can play an important role in terms of building and maintaining efficient and highly liquid financial markets.
On the London Stock Exchange, market makers are called jobbers whereas on the New York Stock Exchange they are now referred to as designated market makers (formerly known as 'specialists').