The Auction House (usually abbreviated as AH by players) is a system that allows you to trade items for gold with other players. You can post your items for sale, and others will be able to search for, and purchase them, from you. Once you've posted your items they will remain on the AH until the auction duration is over, or someone has bought the item.
The AH plays a vital role, particularly concerning professions. It will allow you to get rarer items that are harder to farm, as well as items that you may not be able to acquire yourself, such as materials from gathering professions. It is also a convenient place to buy those items you need to level your professions, for the person with more gold than they have time.
It is also the main avenue to earn gold, either by selling materials you've farmed, items you've crafted or by buying low and selling high. Or, if you are in the market for a WoW token, it is the only place where you can buy or sell the WoW token.