A text link ad is text on a blog or website that is hyperlinked to a specific page on another site. The blog or website that publishes that link (called the publisher) gets paid by an advertiser who is trying to drive traffic to the linked page. Publishers can get paid based on the number of clicks on the text link ad (called pay-per-click) or they could get paid a flat fee for publishing the link.
Text link ads cause some controversy because they have been associated with a drop in Google search rankings or elimination from Google search results (both the publisher and advertiser sites) entirely. To combat the Google penalty, some publishers include the HTML No-Follow tag in text link ads to hide them from Google, but as you can imagine, many advertisers don't like this practice at all.
Popular text link advertising programs include Google AdSense and Kontera, which offer contextual text link advertising opportunities where text on your blog is linked to contextually relevant ad content.