How are parts of Bitcoins traded? I understand how whole Bitcoins are traded (each coin has its own identifier, this identifier is associated with a wallet), though I do not understand how it is possible to trade parts of a Bitcoin (e.g. I can buy 0.7 BTC from Mt. Gox). How are the Bitcoins tracked once they are split into parts, and associated with different wallets?
It is not true that individual bitcoins have an identifier. In fact, individual bitcoins do not exist.

What your wallet holds are addresses. These addresses can occur in the Bitcoin block chain, which can - at its simplest - be seen as a big database of balances for each Bitcoin address. Your wallet also holds a private key for each address, which can be seen as the password needed to spend the balance that is accredited to the corresponding address.

When you spend some bitcoins, you send them from one of your addresses to another address. Only the person that owns the private key corresponding to the address, can spend the bitcoins on its balance.

Now, that still not explains how bitcoins can be divided into smaller fractions. In fact, the Bitcoin protocol does not really work with the unit bitcoin, but with a smaller unit, called satoshi. 1 bitcoin equals 100,000,000 (= 10^8) satoshi. So 1 satoshi (=10^-8 BTC) is the smallest amount you can send.

This means that when you send 0.7 BTC, you actually send 70,000,000 satoshis.