There are several cryptographic technologies that make up the essence of Bitcoin.
First is public key cryptography. Each coin is associated with its current owner's public ECDSA key. When you send some bitcoins to someone, you create a message (transaction), attaching the new owner's public key to this amount of coins, and sign it with your private key. When this transaction is broadcast to the bitcoin network, this lets everyone know that the new owner of these coins in the owner of the new key. Your signature on the message verifies for everyone that the message is authentic. The complete history of transactions is kept by everyone, so anyone can verify who is the current owner of any particular group of coins.
This complete record of transactions is kept in the block chain, which is a sequence of records called blocks. All computers in the network have a copy of the block chain, which they keep updated by passing along new blocks to each other. Each block contains a group of transactions that have been sent since the previous block. In order to preserve the integrity of the block chain, each block in the chain confirms the integrity of the previous one, all the way back to the first one, the genesis block. Record insertion is costly because each block must meet certain requirements that make it difficult to generate a valid block. This way, no party can overwrite previous records by just forking the chain.
Both the chaining, and the difficulty, are achieved via the SHA256 cryptographic hash function. The hash function essentially takes a block of data, and transforms it, in an effectively-impossible to reverse or to predict way, into a large integer. Making the slightest change to a block of data changes its hash unpredictably, so nobody can create a different block of data that gives exactly the same hash. Therefore, by being given a short hash, you can confirm that it matches only a particular long block of data. This way, Bitcoin blocks don't have to contain serial numbers, as blocks can be identified by their hash, which serves the dual purpose of identification as well as integrity verification.
The difficulty factor is achieved by requiring that this integer is below a certain threshold - the data in the block is perturbed by a nonce value, until the data in the block hashes to produce an integer below the threshold - which takes a lot of processing power. This low hash value for the block serves as an easily-verifiable proof of work - every node on the network can instantly verify that the block meets the required criteria.
With this framework, we are able to achieve the essential functions of the Bitcoin system. We have verifiable ownership of bitcoins, and a distributed database of all transactions, which prevents double spending.
We have mentioned in the previous section that adding a block to the block chain is difficult, requiring time and processing power to accomplish. So what incentive does anyone have to spend the effort to produce a block, if it takes up all these resources? The answer is that the person who manages to produce a block gets a reward. This reward is two-fold. First, the block producer gets a bounty of some number of bitcoins, which is agreed-upon by the network. (Currently this bounty is 50 bitcoins; this value will halve every 210,000 blocks.) Second, any transaction fees that may be present in the transactions included in the block, get claimed by the block producer.
This gives rise to the activity known as "bitcoin mining" - using processing power to try to produce a valid block, and as a result 'mine' some bitcoins. The network rules are such that the difficulty is adjusted to keep block production to approximately 1 block per 10 minutes. Thus, the more miners engage in the mining activity, the more difficult it becomes for each individual miner to produce a block. The higher the total difficulty, the harder it is for an attacker to overwrite the tip of the block chain with his own blocks (which enables him to double-spend his coins. See the weaknesses page for more details).
Besides being important for maintaining the transaction database, mining is also the mechanism by which bitcoins get created and distributed among the people in the bitcoin economy. The network rules are such that over the next hundred years, give or take a few decades, a total of 21 million bitcoins will be created. Rather than dropping money out of a helicopter, the bitcoins are awarded to those who contribute to the network by creating blocks in the block chain.