It's the text that shows up in a live link.
For example - in the link below, the anchor text is “Friday Traffic Report”:
Friday Traffic Report
Good search engine optimization strategy includes choosing your anchor text carefully. Anchor text is an important factor in how search engines figure the relevance of your content.
Use your article's important keywords (that are relevant to whatever the link leads to) wherever possible in your anchor text.
Anchor text is valuable and should be used wisely. Avoid wasting an anchor text opportunity on unimportant words like “click here”, “this page”, etc.
Always think in terms of optimizing your anchor text for relevant, useful keywords.
It's worth taking an extra moment or two to decide the best way to word your sentence so you can use the best possible anchor text in a link.