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 is 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.
To make generating bitcoins difficult the Hashcash cost-function is used. Hashcash is the first secure efficiently verifiable cost-function or proof-of-work function. The beauty of hashcash is that is is non-interactive and has no secret keys that have to be managed by a central server or relying party; hashcash is as a result fully distributed and infinitely scalable. (Hashcash uses symmetric key cryptogaphy, namely a one-way hashcash function - typically either SHA1 or SHA-256).
In bitcoin, integrity, block-chaining, and the hashcash cost-function all use SHA256 as the underlying cryptographic hash function.
A cryptographic hash function essentially takes input data which can be of practically any size, and transforms it, in an effectively-impossible to reverse or to predict way, into a relatively compact string (in the case of SHA-256 the hash is 32 bytes). Making the slightest change to the input 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 compact hash, you can confirm that it matches only a particular input datum, and in bitcoin the input data being a block-chain is significantly larger than the SHA-256 hash. 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. An identification string that also provides its own integrity is called a self-certifying identifier.
The hashcash difficulty factor is achieved by requiring that the hash output has a number of leading zeros. Technically, to allow more fine-grained control than Hashcash number of leading 0-bits method, Bitcoin extends the hashcash solution definition by treating the hash as a large big-endian integer, and checking that the integer is below a certain threshold. The hashcash cost-function iterates by perturbing data in the block 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.
