What is the function of the html "rel" attribute?

Typically they are used to provide information to search engines about the structure of your website. For instance, you can give links a next or prev rel attribute for paginating links (links which show the next/last set of search results), or you could supply a canonical value to tell search engines which is the default link for the page it is looking at (sometimes pages are accessible via a number of different links, and this avoids indexing of duplicate content which could hurt your SEO).

Here are couple of links from Google regarding best usage:

https://support.google.com/webmasters/answer/1663744

https://support.google.com/webmasters/answer/139394