What are bitcoin "confirmations"? When I receive bitcoins I see a certain number of "confirmations" in the client. What do these mean? Why do most sites make me wait for 6 "confirmations" before a purchase will go through?
Bitcoin confirmations represent the number of blocks in the block chain that have been accepted by the network since the block that includes the transaction.

In simpler terms it represents the difficulty of a double spend attack. With zero confirmations no proof of work has been done, so you can't tell if anyone considers the transaction valid.  Even with a single transaction it is possible for an attacker to pre compute a single block. 

With zero or even one confirmation a double spend is very possible since the next block that is solved may confirm a different block instead of the one that has the transaction.  That different block my show the coins being spent elsewhere.  The odds that a double spend has occurred gets exponentially smaller with each confirmation.

An attacker must match the power of the entire bitcoin network to keep up with block creation so as time goes by it becomes increasingly difficult to forge a transaction.

It is generally accepted for most transactions that 6 confirmations represent enough security to assure the transition is valid.