http://beginnersinvest.about.com/library/blmarketmakers.htm
Market Makers are Very large financial firms that are specialists in particular stocks. For the privilege of being a market maker, they must abide by certain rules, one of which is that they must always give a Bid and an Ask (Buy & Sell) price in the stocks they deal in. Thus they "Make The Market".
Market Makers need to have a vast supply of money to be able to operate as a MM.
The link above will give you a bit more info on them in ordinary language as well as a few links to other stock market questions.
Some of the following are Market Makers on the Nasdaq:
GOLDMAN, SACHS & CO
J.P. Morgan Securities Inc.
Citigroup Global Markets Inc.
MORGAN STANLEY & CO., INCORPORATED
Credit Suisse First Boston LLC
Banc of America Securities LLC
I am not being cheeky, but you are not likely to 'become' a market maker. You may go and work for one of them, but the average person doesn't become one.
I trade the Nasdaq, so can really on comment on that. But, with direct access into the nasdaq, you can create your own bid and ask prices, thus almost being a market maker yourself. I say almost as that is the very tip of the iceberg of their job as a MM.