If I open Google and type 1 Bitcoin to Dollar I'm returned with a number (as of writing, "1 Bitcoin equals 246.09 US Dollar"). But where does this number come from?
Given the decentralized nature of Bitcoins (there are no banks, governments or whatsoever), shouldn't each exchange have its own rate? How can there be a unique, widely-accepted exchange rate, valid for everyone?
When exchange 'A' sells bitcoins for 200 dollars and exchange 'B' buys bitcoins for 300 dollars, there will be people who buy at 'A' and sell at 'B' as long as they can make money on it. This is called 'arbitrage'.
Because of the open markets, people are free to do as they please, so there are people who buy and sell solely to make mere cents on the dollar. Because of this, there is usually not a lot of difference between the exchanges. The only difference in price between the exchanges as we can see them right now, is because of trading and/or withdrawal fees. (Exchange 'A' is 1 dollar cheaper than exchange 'B', because exchange A requires 1 dollar per withdrawal, meaning in the end you end up paying the same amount for 1 bitcoin at both exchanges.)
Because of arbitrage, Google can pretty safely say the price of bitcoin is about XXX dollars / euros because if one exchange would go up or down with the price, they all would. (Exceptions sometimes occur, when one 'big fish' / whale dumps a lot of bitcoins on 1 exchange only, crashing the price on exchange 'A', but not immediately on exchange 'B'. Of course, because of arbitrage, this difference will become smaller with time.) Therefor there is no need to let a user know which exchange offers the lowest price, because they will most likely offer bitcoins for about the same price anyway.
Also, exchanges do not 'set' an exchange rate, they merely offer people with bitcoins the opportunity to sell them to people looking to buy bitcoins. If everyone would suddenly start asking 1000 dollars per bitcoin (even though right now they're worth $245 each), there would be no other choice than to buy bitcoins at 1000 dollars each. Other than that, there is no way of 'setting' the price. (This price is never set by the exchange, only by the sellers.)
Technically speaking; Google gets the number by using the API (Application Programming Interface) of an exchange. Exchanges such as Kraken offer API's from which software can request the latest exchange rates. This way Google can let its users know what the latest exchange rate is.
People buy low and sell high across exchanges.
It keeps bitcoin prices from different exchanges stay almost the same.
In early days, the difference among exchanges could be 50 $/bitcion. It might be still profitable to catch these opportunities : )
Here is a orderbook from different bitcoin exchanges when I am writing answer
