It took me some time to understand what BitCoin actually is.
Basically, BitCoin network is a big transaction database.
One transaction is: "transfer X amount of BitCoins from account Y to account Z." This 'database', or transaction log is replicated and stored on all participating users' computers.
You can be sure a (your) transaction has been recorded, because you can check with many other peers who will verify that it is.
Of course, the inner parts are more complex, and there's a way to generate new BitCoins (but over time you can generate less and less, so it's a finite amount in total).
(I forgot to login before my comment, so i'm posting it again.)