I understand that rel got something to do with relationships between things (documents? elements?) but I don't really get anything past that. What exactly does it do when used in the achor tag ? Also, are there specific values for x in rel="x"?
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); A nofollow to inform search engines not to crawl (this is also good for not passing SEO 'link juice' to external or low priority pages); 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).
This describes only a few possible uses - it is a very versatile tag.
With regards to navigation, pagination and canonicalization, here are couple of useful links from Google:
https://web.archive.org/web/20180125083221/https://support.google.com/webmasters/answer/1663744