So, you have a lot of items. Some of them you'll want to use or meld, but there are items you may not have any use for, and items that you really want but you don't have. Let's examine the trading system.
The currency for all transactions in the game is the coin. There is no direct player-to-player trading system; the only way to exchange items is to sell something on the market for coins, and use the coins to buy the item you want. You can see the number of coins you have by looking at the green number to the right of perfstats.
So, you have an item and you want to get rid of it. Open your inventory, and click on that item. You'll get a menu that looks like this:
The liquidate button will sell your item to the person with the current highest bid for that item, which in this case is three coins. If there is no buy offer for the item, the liquidate button will be replaced with a delete button, and you won't get anything for the item.
If you're not satisfied with the current buy offer, you can click "List for Sale" to specify a price. The item will remain where it currently is (in particular, if your item is in short term storage and expires before someone buys it, you won't get your coins). You can also see the price history chart of that item, allowing you to estimate a fair market value for it.
If there is an existing buy offer for the item you're selling higher than the listed price, the transaction will go through automatically.
As with selling, there are two ways to buy stuff.
You can browse the items for sale catalog by clicking on the red icon on the left of the screen. That will bring up a screen listing items for sale and their prices.
If you see something you like, click on it. You'll get a purchase confirmation screen with a graph of how much this item has sold for in the past. If you want the item and the price looks right, click OK to purchase it.
If you don't see the item you want, or the price is too high (as is often the case), put out a buy offer for the item by clicking on the green icon on the left side of the screen, and clicking "New Buy Offer". You'll see something like this:
You can specify the item type, level, number of copies you want, and the price that you're willing to pay. The buy offer will be posted once you click OK, and if your buy offer is the highest (which you can check by looking at the current high buy offer), you'll get the item when someone liquidates it.
Note that buy offers are normally absurdly low and prices in the shop are normally absurdly high. The prices that items typically sell for is somewhere in between. Don't buy or sell anything before checking the price graph to make sure that you're getting a reasonable deal.
Often you'll want to give someone an item, or someone will want to give you an item. Since player-to-player trading is not possible, the procedure for that is for the receiver to post a buy offer marginally greater than the current highest buy offer and for the giver to liquidate the item.