Since this difficult process is different for every country, I'll list what I know.
USA
Priceline is my favourite website for flights. When you can fly from LAX to New York for $1 + $35 taxes, you know you've got yourself a deal. I didn't actually book that flight, as I didn't actually need to cross the US that day, but I'm sure you can get a cheap fare like that. At priceline.com there is a box below the main booking interface called "Name your own price". This is where you choose how much you want to pay! Having trouble? Bidding for Travel can help you out.
United Kingdom
In the UK it's usually just best to go straight to the airline. You've heard it before, and I'll stress it again: RyanAir and EasyJet are the cheapest for European flights. I stumbled upon a deal today... free flights. Nothing to pay. Not even taxes (they're included). That's on RyanAir. Wow - I wonder how they can be in the top 3 most profitable airlines in the world when they "sell" tickets like that?
Australia
Australia is one of the most expensive places to fly in, distance-wise, and there aren't really that many super-cheap sales. However, Jetstar gives tickets about AU$200 less than the other airlines. It's budget - but once you factor in food and everything you're still getting a huge discount. Jetstar also has on-demand in-flight entertainment - something the other domestic airlines don't have. So if you're willing to connect in Melbourne Airport and good at lining up times, give Jetstar a go. If not, try Virgin Blue.
