This article in The Independent contains the best real-world analogy that I’ve seen for how AV works from the voter’s point-of-view:
Scenario one: You hand over £1 for a bag of crisps and tell your friend you want a packet of salt and vinegar. But when he gets to the shops, it turns out there are none in stock. So your friend asks the shopkeeper what’s popular, he comes back with prawn crackers and you end up with someone else’s choice.
Scenario two: You hand over £1 for a bag of crisps and tell your friend you want salt and vinegar. If not, your second preference would be cheese and onion; failing that, ready salted. Your friend/butler toddles off to the shops and returns with cheese and onion. You have paid the same money but got something closer to your ideal outcome.
Scenario 1 is how the current voting system treats our votes, and scenario 2 is how AV treats our votes.