BoardGameGeek provides several ways to sell or trade boardgames online.
The first is to simply list it as available for trade. Once you have an account on BoardGameGeek, at the bottom of every game page you can add the game to your collection (click on the "user info" link at the top to get there quickly; for example). You can then click "add a copy to your collection" to list the fact that you own the game, and then fill in information on it; check boxes to indicate that you're willing to trade the game, fill in information on the condition of the game, etc. For games that you would like in trade, again go to the page for the game, go down to the "user info", click "record information", and check the "want in trade" box.
Once you've filled in some games you have available for trade, and want in trade, you can try and find trades. Go to the "My Geek" menu at the top, and select the "Trades" option. This will bring you to the Trade Manager, where you can find potential trades and track the process of trades in progress. Click "Direct Matches" to find people who have games for trade that you want, and who want games that you have for trade. You'll likely get a long list of matches, telling you what games they want and what games they have. You can then propose a trade with them, which will send them a GeekMail proposing the trade, that they can accept or decline. If they accept, you work out the shipping details with them, and then send each other the games. If you leave games listed as available for trade, other people who find them through the trade finder may also initiate trades with you, so if you're hoping to trade some games, check back relatively often to see if you get any GeekMail with trade offers.
There are also other ways to trade games via BGG. Some of the popular ones are "ultimate trades" and "math trades". The principle behind both is the same, to try and create circular trades, though the mechanism is different. The idea is that if you have game A and want game B, you might not be able to find someone who has game B and wants game A; but there might be one person who has game B and wants game C, and another who has C and wants A, so you can do a circular trade, where you send A to the person who wants it, he sends C to the person who wants it, and that person sends B to you.
Ultimate trades are generally posted as a GeekList, where someone starts the trade off by listing a game that he has for trade (so, it will start off as a list of one item). Then anyone who wants that game will come along, and post a new game that they would offer up for the original game (the new game should be of roughly equal value, or if it's a bit cheaper than you might also throw in a "sweetener" of a cheap game to even it out). This goes on, with each person in turn saying that they'll be willing to take the last item in exchange for a new one that they post. Eventually, the person who started the trade will see a game he likes, and close the loop by saying that he'll take that game. Then everyone shares shipping information with the person before them in the list, and sends their games to the people who claimed them. You can find examples of ultimate trades by searching for such GeekLists.
Math trades are a generalization of ultimate trades. The problem with ultimate trades is that they'll sometimes lose steam before you complete a loop, and so you will never complete them; or sometimes people have games to trade but don't want to initiate an ultimate trade nor do they find an existing, open one with games that they want. In a math trade, each person lists one or more games that they have for trade. Once they've done so, a list of games available is compiled, and everyone lists, for each game they have available for trades, one or more games from the list that they would like in exchange. Once all of this information has been submitted, someone runs one of several tools that will crunch all of the numbers, and try and find a set of trades that will maximize the total number of trades completed. It may find one big circular trade, but more likely it will find several smaller cycles, that satisfy more people than one big circular trade would have been able to. Then you all exchange shipping information, and ship the games. You can find examples of math trades by searching for them.
All of these trading methods allow you to leave feedback on the reliability of the person you're trading with, just like eBay. I've only done a few trades through BGG, but I've found everyone that I've dealt with honest and prompt about shipping their games.
Finally, there is the BoardGameGeek Marketplace, where you can buy and sell games. If you list games for sale on there, they will appear on each game page, so people interested in the game can buy them quickly and easily. I have never used the marketplace, so I can't speak about it much, but you can find some more info on it on the BoardGameGeek wiki.
And not exactly trade related, but similar, are the Secret Santa or other gift exchanges that go on. These are usually intended for new games (thus not directly relevant for trading purposes), but they are good ways for sharing your love for games with others. They will frequently be started as threads listing the rules for the exchange (the target price point for the games, any theme or other restrictions), and people can sign up to be given targets. It used to be run manually through GeekMail, but now there is a tool for creating and browsing gift exchanges, which can be private or public.