The block chain is a common ledger shared by all Bitcoin nodes which details the owner of each bitcoin, or fraction thereof. Unlike conventional banking systems, there is no central place where this ledger of transactions is stored. This is accomplished through the broadcasting of small pieces ("blocks"), each stating that it is a continuation of a previous block. It is possible for the block chain to split; that is, it is possible for two blocks to both point to the same parent block and contain some, but not all, of the same transactions. When this happens, each computer in the network must decide for itself which branch is the "correct" one that should be accepted and extended further.
The rule in this case is to accept the "longest" valid branch. Choose from the branches of blocks that you have received, the path, the total "difficulty" of which is the highest. This is the sequence of blocks that is assumed to have required the most work (CPU time) to generate. For Bitcoin, this will be the "true" order of events, and this is what it will take into account when calculating the balance to show to the user.
It is still possible that, as new blocks are constantly being generated, at some later time, some other branch will become the longest branch. However, it takes significant effort to extend a branch, and nodes work to extend the branch that they have received and accepted (which is normally the longest one). So, the longer this branch becomes compared to the second-longest branch, the more effort it will take for the second-longest branch to catch up and overcome the first in length. Also, the more nodes in the network hear about the longest branch, the more unlikely it becomes for other branches to be extended the next time a block is generated, since the nodes will accept the longest chain.
Therefore, the more time a transaction has been part of the longest block chain, the more likely it is to remain part of the chain indefinitely. This is what makes transactions non-reversible and this is what prevents people from double-spending their coins. What the receiver of each transaction does, after money has been supposedly transferred to him/her is to check how long the block chain following the said transaction has become, because the more blocks are added to the longest branch after the transaction, the less likely is it that some other branch will overcome it.
When the block chain after the transaction has become long enough, it becomes near-impossible for another branch to overcome it, and so people can start accepting the transaction as true. This is why 'blocks' also serve as 'confirmations' for a transaction. Even if another branch does overcome the one with the transaction, most of the blocks will have been generated by people who have no affiliation with the sender of the coins, as a large number of people are working to generate blocks. Since transactions are broadcasted to all nodes in the network, these blocks are just as likely to contain the transaction as the blocks in the previously-accepted branch.
Bitcoin relies on the fact that no single entity can control most of the CPU power on the network for any significant length of time, since, if they could, they would be able to extend any branch of the tree they chose, and faster than any other branch can be extended, making it the longest branch, and then permanently controlling which transactions appear in it.
