Bitcoin Exchanges are market places where users can buy and sell bitcoin (and in some cases other digital currencies) with their local fiat currency. In a lot of ways, they function very similarly to Stock Exchanges though technically they are more closely aligned to a Foreign Exchange Spot Market at a transaction level.
Trading and Transaction Fees
The trading platforms have a central limit order book that consists of bids and offers of different quantities and prices that trade on a time priority basis. Buyers and sellers pay fees ranging from 0% to .25% based on the business model of the exchange.
Some exchanges have a maker-taker model that charges customers based on whether they provide liquidity (rest in the order book) or remove liquidity (trade against a resting order). Other exchanges have a volume-based model where you pay based on the number of bitcoin traded over a period of time.
Clearing and Settlement
In Equities, trades are cleared and settled post trade execution on a T+3 (3 days from day of trade) basis. Most of these trades are centrally cleared and settled by the Depository Trust & Clearing Corporation.
One of the most powerful features about Bitcoin is the blockchain. The blockchain is essentially a distributed database that contains the complete transaction history from the inception of Bitcoin. Transactions are secured, verified, and added to the blockchain by armies of computers that solve complex cryptographic math functions.
Anyone with internet access can view the blockchain, confirm a transaction, and prove ownership. This is exactly how clearing and settlement works in today's financial system. So, there is no need for a central organization to clear and settle bitcoin transactions. This makes trading bitcoin cheaper and faster than trading any other asset class.
Please note that some Bitcoin Exchanges internalize transactions (a.k.a off-line or off-chain transactions) until the need to register the transaction on the blockchain. This introduces a certain level of risk and requires that users trust the exchange with their bitcoins.
Bitfinex is currently the largest bitcoin exchange by volume. There are a number of Bitcoin Exchanges globally, and Tradeblock has a great site that displays the largest, most popular exchanges along with their top of book bids and offers.