Anchor text is the text part of a hyperlink which the user can click to reach another web page. Anchor text is usually – and advisably for good SEO and UX practice – descriptive of the page it's linking to.
The HTML code in the hyperlink gives the url of where the link directs and is invisible to the user. The anchor text, however, is visible as part of the text on the page and usually gives the user some idea of where the link will lead them.
Here's an example of the HTML code for a hyperlink, the anchor text is highlighted in green:
<a href="https://www.example-url.com">example anchor text</a>
Anchor text appears most often blue indicating to the user that it can be clicked on. For example, this is anchor text leading to the Moz blog's excellent guide on the subject.
For SEO benefits it's best to use anchor text which is indicative of the page the link leads to. Most SEOs also advise using keywords in anchor text, although do this sparingly as Google is clamping down after recent black hat activity from link farms and paid link schemes.
A good rule is to use diverse anchor text which is useful to the user. Where possible, you should avoid using anchor text which is lacking description, such as "click here" and "visit this website", as both the user and search engine will have difficulty contextualising this.