Anchor Text: Anchor text is the part of a hyperlink that's clickable. It's the word, phrase, or sentence that's usually underlined and/or colored differently from the normal text, alerting the reader that it's a link to another page.
For example, popular news blog Gawker published a story titled "Apparently Kid Plays With Puppies in His First Commercial, Apparently." In the first line of the article, the phrase "Apparently Kid" is orange instead of black, signifying that it's a link. Clicking on it sends the reader to a previous post on Gawker that details who Internet sensation Apparently Kid is.
Naturally, anchor text is a crucial part of Search Engine Optimization (SEO). As search engine algorithms have gotten more complex and mature over time, they've added more metrics to determine the quality, authority, and relevancy of sites. One of the most important metrics that's been added is link relevancy, which is determined by both the source page's content, and -- you guessed -- the anchor text.
To better understand what link relevancy is and its importance to SEO, let's return to our previous example. Gawker chose to make their anchor text the phrase "Apparently Kid," which was relevant not only to the article being published, but to the article being linked to as well. To make it more relevant, they might have made the anchor text a phrase like "who Apparently Kid is," or "where Apparently Kid came from."
Such choices are essential to search engines, which use the anchor text and the link itself to determine what the target page is about so that it can match it and the article with the link to appropriate search queries.
However, people took advantage of the SEO potential of links, and stuck links in to places where they wouldn't naturally go just to make other pages rank better. As a result, Google put out the Penguin update, which penalized those who were trying to use links to manipulate rankings, and made it all the more important for anchor text to be relevant.
Essentially, anchor text is the clickable part of a link that gives both humans and search engines a general idea of where it goes.