Bitcoin currency exchanges work in a manner similar to banks. One first deposits amounts of money in the currencies supported by the exchange, to his own account in the exchange, uses these balances to trade with other users of the exchange and then withdraws that money. Unlike over-the-counter transactions, there is no risk of losing money due to people not fulfilling their part of the deal, as long as the exchange itself does not commit fraud or withhold money.
Exchanging is done by placing "buy" or "sell" orders, which the exchange system software then matches with each other. "Buy" orders (or "bids") are offers to buy bitcoins in exchange for another currency at a maximum price-per-bitcoin which is set by the offerer. "Sell" orders (or "asks") are offers to sell bitcoins at a minimum price-per-bitcoin. If the bid price of a buy order is higher than the ask price of a sell order, an exchange can be performed and either the bid order, the sell order or both can be removed from the "order book". Thus, at any given time, there is a price above which there are no more buy orders and a slightly higher price below which there are no more sell orders.
Communication with the Bitcoin currency exchanges is commonly done using a standard web browser, over a secure SSL connection.
"Soft currencies" and chargebacks
Exchanging bitcoins for other forms of currency brings up some issues regarding chargeback fraud. Specifically, payment methods such as credit cards, and PayPal, can be reversed up to 90 days after the transaction took place. In contrast, bitcoin is a "hard currency", once you spend bitcoins, you cannot get them back by 'pulling' from your side. Thus, when you trade bitcoin for a 'soft' currency like paypal or credit card, you open yourself up to the risk of chargeback after you send bitcoin. The buyer may initiate a chargeback by claiming non-receipt of goods, or if a stolen account was used, the real account owner will initiate the process once he notices a charge he didn't make. As a result, it is strongly recommended to not trade 'soft' currency for 'hard' currency with people you do not know or trust.
The two major exchanges at the time, Bitcoin Market and MtGox, were hit with a wave of PayPal scams in October 2010, where one or a group of individuals used stolen PayPal accounts to fund their exchange accounts to buy bitcoins. This has caused the freezing of the Mtgox paypal account, and a suspension of new user registration on Bitcoin Market. These account freezes caused a temporary liquidity problem for the bitcoin economy, as it became more difficult to exchange dollars for bitcoins.
Exchange Rates and Market Forces
Early in the life of Bitcoin, the currency showed some major fluctuations in exchange value, ranging from under $50 to $266 US. The exchange rate of Bitcoin has shown relatively stable growth since the beginning of 2013. According to Currency.Wiki , as of February 01, 2014, the current exchange rate for bitcoins is at $959.58 US. The all time high for the value of a single bitcoin was on November 17th, 2013 when it reached $1216.73 US on the Mt. Gox exchange.
Bitcoin has been criticized by economists for bubbling up around itself, similar to the housing market in the US before the crash and it is true that Bitcoin has shown a tendency for rapid rises and crashes in price. However, given the instability of the global economy, Bitcoin has proven to be a reliable investment compared to many other popular currencies. In particular the European debt crisis gave rise to a large amount of currency being converted to bitcoin to keep it safe from the falling value of the Euro. These investments in turn drove up the value of the bitcoin thanks to its unique production method.
Despite the growing popularity of bitcoin transactions and the generally rising price, several events have shown an inability to withstand major blows to its reputation. The FBI seizure of around 26,000 bitcoins from the drug distribution website Silk Road more than halved the value of individual bitcoins. Technical errors have also proven to be a difficulty for speculators on the exchange value of the BTC. In April of 2013, a backlog of transactions shut down the Mt. Gox website, causing a drop in value from $266 to $77 US.
While it has become easier to buy and sell bitcoins with the influx of bitcoin exchanges  across the world over the past couple of years, the availability of access to the currency has been one of the greatest sources of variability in its exchange rate. Because of looming government regulations, most notably the U.S., it has been a challenge to convert your local currency into Bitcoin, which has led to highly variable prices based on geography, which, for a currency which is designed to be borderless, has proven to be an issue, given that so many bitcoin users treat their bitcoins more like an asset than a currency. That is to say, they intend to hold onto their BTC until they decide to convert back into their local currency, rather than spend it online for anonymous transactions.