Auction Houses are places where you can go and sell/purchase items to/from other players. There are two Auction Houses in every city. Items from these houses are listed across all the continents, which means that items that are plentiful in one area and scarce in another can be exported quite nicely (hey! this here is economics!)
When you sell an item at the AH, you pick a price for it to sell at. Someone else somewhere else "bids" on that item. This isn't really bidding in the strictest sense. Basically, they pick a price, and if the price is equal to or exceeds your price, and is still less than others selling the same item, they purchase your item for the price they bid at.
Auction Houses can also be one of the quickest ways to make money.
How do I make money at the Auction House?
Ooo... now there's the 50 million dollar question ;-)
Essentially, it is just like any other commodities market (which is what the AH is). You need to buy low and sell high. Buying low may mean purchasing something from the AH, from an in-town store, obtaining it through a quest, or finding it in the wild. Selling high means you sell it enough to make it worth your effort or money.
The trick is to beat the other sellers around you. Whenever you go to sell an item, you will see how many others are selling the same item as you. You will also be able to see the recent sales history. If the item you are selling has been selling for 300 gil, and there are currently 20 other people trying to sell it, then trying to sell yours for 350 gil is probably not going to work (because, chances are, everyone else around you is selling it for less).
What I have found works is to pick one of the lower recent bids, and sell it for a certain number of gils less. For example, if an item has been selling for 600 gil, I may place my price at 558, ensuring that I would be the next one to sell as long as my price is lower than my competitors. However, usually only the newiest of newbies don't know this trick, so you have no real garantee that others aree selling at the recent sales prices. Likewise, a lot of people bidding know this, and will use it to their advantage to pinch pennies and drive the price lower.
This can lead to a problem where the value of an object plummets as the sellers try to beat their competitors, which is obviously good for buyers, but not for the sellers. Often when this happens, a few brave souls tend to try to sell high again to bring up the commodities price.
Be just a bit careful with that FAQ, because some of the things it says (not many) are wrong (for example, they say that AH recent sales vary from city to city on the same server, this is, in fact, not true. AH recent sales are shared across certain key cities).