Hi! Welcome aboard!

Have you read the FAQ yet? Among other things it has a link to the Bitcoin whitepaper. The minutiae of the protocol and inner workings are, unfortunately, only "documented" in the form of the application's source code.

The basic idea is that coins are signed cryptographically, so you can only spend coins you actually own. To spend a coin you sign it over to someone else using their public key -- the address is a signature of that key and serves to identify it.

To prevent you from making a copy of a coin and spend it twice, the clients form a P2P network that records which coins have already been spent in a long chain that is linked by hashes (see the paper). The IRC channel merely is a relatively simple way to get the IP addresses of a few other participants, so you can join the network.

Apart from recording all transactions, the hash-chain also serves as a sort of distributed clock that makes sure you can't alter past transactions or create many new, bogus ones to DoS the system. To make a new block valid, the clients have to twiddle a nonce inside a block until the block's SHA-256 hash is below a target number. The target number is adjusted about once every two weeks to adapt the difficulty of generating a valid block to the computing power of the network -- basically this ensures that a valid block of transactions is generated about six times per hour, no matter how many or few people are participating.