Linking is a big part of your SEO strategy. Search engines can both see and acknowledge the links you put on your site, along with links to your site from other sources. While it sees them all, it only acknowledges or recognizes the dofollow links. For those links with the added nofollow tag, it ignores the link.
At least, that's how it worked until last month.
As a quick overview, most links on a webpage, with typical HTML coding, are dofollow links. That's the standard way that a search engine reads the code. It only changes if you manually add in a "nofollow" tag to that code.