The heart of the Bitcoin network is a shared, public record of Bitcoin transactions known as the blockchain. Every Bitcoin transaction that has ever occurred is listed in the blockchain, and every node (i.e. computer) in the Bitcoin network has its own copy. The blockchain is organized as a list of blocks, each of which contains transactions that occurred during a particular period of time.
When a Bitcoin user wants to make a transaction, she announces the transaction — the sender, recipient, amount, and other information — to the network. Nodes share these announcements in peer-to-peer fashion so that everyone soon knows about them.
Nodes check that each transaction complies with the rules of the Bitcoin system, combine all the valid transactions they have heard about into a block, and then work to solve a difficult mathematical problem that takes this block as an input. The first node to solve the problem announces the solution to the others. The other nodes verify that the problem was solved correctly. If it was, then they put the block at the end of their copy of the blockchain. Once most nodes have done this, the block is considered to be an official part of the blockchain, and transactions in that block are considered final.