This is has really taken off since Google started AdWords. Your text ads appear alongside search results. Where you show on the page depends on how much you are willing to pay. You only pay when someone clicks on your ad!
This where you and another site agree to link to each other. Don't do it! You might gain in the short-term, but in the long-term you run a serious risk of your site being penalized by search engines
They all work the same way - you display a banner ad on your site (delivered automatically from the exchange), and for every X visitors to that page, you will get your banner ad shown Y times on the sites of others. The ratio varies from 2:1 to 5:4, with the remainder being paid advertising. If you already have traffic, the ads on your site can make it look cheap (most ads are designed by amateurs). If you don't have any traffic, you won't get your ads displayed elsewhere. The only scenario where this might pay off is where you have a very sticky site (perhaps a forum) with lots of returning visitors. Otherwise AVOID.