Characters unlock the auction once they reach level 20. From that moment they can browse currently auctioned items at any moment by pressing the R key; but in order to sell something, you need to find an auction employee. They can be found in the major cities: Sidus Ur, Sulan, and Fort Whetstone. You can use the NPC search option to make it easier: just open the world map, click the magnifying glass icon at the top right and type "auction" in the pop-up.
Additionally, when characters reach level 40, they can unlock the Quick Auction option, this allows them to use it in full from wherever. You will have to unlock the fourth chapter of the Artisan Manual in the third volume of the Life Skill Book. Once done, you will be able to sell and buy any items you are interested in by pressing R.
The filter and sorting system will make the search easier. You can set the level range of the item you are looking for, the amount you need, sort the lots by various attributes, compare prices for individual items and for various combinations, etc. There is a separate tab for monitoring lots you have already bid on. Also, items with bids are marked with a special round symbol. If you find such an item, you can point at it and see the name of the character that has bid on it. All in all, you will be provided with the whole range of tools to make the system easy to use.
You can put items up for sale for 24, 48, and 72 hour periods. Selling an item incurs a storage charge of 1% of the purchase price, but no less than 10 and no more than 200 Gythil. So remember to keep an eye on your funds before starting the trade. Additionally, the auction withholds a fee of 5% from the price of the item.