On a Bitcoin exchange, the price of a Bitcoin is not set by exchange itself. The exchange provides a marketplace where someone who wants to buy bitcoin can place a bid at the price they are willing to pay, and when another person puts it up for sale at the same price, the trade is executed. It's a bit like buying shares of a stock. The prices set for these exchange trades are what determine the value of a Bitcoin. Using an exchange means you are actively participating in the Bitcoin free market, and many offer advanced options for people who want to "trade" Bitcoin as an investment- but they do have stricter proof-of-identity requirements due to anti-money laundering regulations.
Here are a few well known examples:
Bitfinex - This exchange offers trading of Bitcoin as well as Litecoin. It is one of the first to offer advanced trading features such as margin trading and liquidity swaps. It uses the "maker taker" model, and charges fees based on how frequently you trade.
BitStamp - As one of the most popular exchanges in the world, BitStamp has grown to prominence as a reliable exchange to buy and sell Bitcoin. New users submit documents (copy of photo ID, proof of residency) which takes 1-2 days to verify new accounts. Users then make bank wire transfers to deposit funds onto the exchange platform, which can take 2-5 days. Once funds are in the account, users can buy and sell Bitcoin between users. Trading fees start at 0.5% and decreases with higher volume. There is also a small fee to withdraw funds from the exchange that varies from country to country.
BTCChina – By sheer trading volume that takes place on Bitcoin exchanges, BTCChina controls 70 percent of the market share. Although the Bank of China restricts renminbi (Chinese currency) deposits and withdrawals from Chinese Bitcoin-related companies, BTCChina has found workaround solutions for its domestic traders, while managing to scale internationally by launching USD trading accounts. Users are subject to a rigorous sign-up process before being allowed to buy and sell Bitcoins. Once the account is set up and fully verified, trades are free, and withdrawal fees are 0.38%.