From Bitcoin Step by Step:
Trading BTC
Let us start with how BTC is exchanged. In simple terms, the process of sending BTC from one user to another is based on asymmetric cryptography. This means that if Alice wants to send Bob 12.5 BTC, the following would have to occur:
1. Bob uses the wallet application to create an address, which is represented by the software as a long string of alphanumeric characters.
2. Bob gives Alice the address which is the public key for which Bob has a matching private key in his wallet.
3. Alice uses her Bitcoin application to send 12.5 BTC to Bob using the provided address.
4. The transaction is made public on the network and miners begin to process it.
5. Once the transaction has gone through enough cycles on the network, it is released to Bob.
The 12.5 BTC are now no longer Alice’s. The 12.5 BTC now belong to Bob. The information that proves Bob’s ownership exists in the database of blocks which is stored on your local PC by the wallet application or on the online wallet service if you choose an online wallet. The proof of ownership for Bob is that private key which is stored in his wallet. The fact that everyone in the network has the same information protects Bob’s right to ownership as long as he retains the secret keys. He could give someone his wallet, or it could be stolen from him and the bearer would have the ability to take the BTC and transfer them to another account.
I'd quote it, but its my book.
I didn't go into how change is handled, the book was geared for Newbies.