Spiders from search engiines "crawl" the web by following a link from one page to another. Using a rel=nofollow HTML attribute on a link tells the search engine not to follw the link and crawl the page that the link points to.
Search engine rankings by sites like Google are based on the premise that the best content has the most links to it (this is a super over-simplification, I know). "nofollow" links aren't counted by Google when they assign value to the page being linked to.
Most often sites use "nofollow" to block links from user generated content that might be used to manipulate search engines. An example of this might be a blog that nofollows all the links in it's comments so that spammers don't show up and flood the comment section with links to sites they'd like to rank higher in google.