How is the exchange rate for Bitcoin established? I noticed that there are exchange rates for Bitcoin to other given currencies. How are these rates established?
You have people that have BTC or currency already in their accounts.
Those people can claim: "I will buy/sell X amount of Bitcoins for Y amount of currency". If you divide one by the other, you have the exchange rate for that particular transaction.
If a buy and sell transaction overlap, they are executed and removed from the system.
After all such transactions are removed, you have a gap between the exchange rate at which people want to buy Bitcoins, and that at which people want to sell Bitcoins.
Average the closest buy-sell pair of transactions and you have the exchange rate, halfway between those two, or the price of the last trade may be used. In both cases, it's only an estimate of the price that the next trade will take place at.