'No follow' links tell web crawlers not to check the link, while 'do follow' tell web crawlers to follow through to the link.
An example would be posting a link on a blog that's relevant to your content in order to get 'juice' from Google or another search engine. By making the links in a blog 'no follow', when Google indexes the page (or subsequently rechecks the page for updated content - 'crawling'), Google does not follow through to your link and no 'juice' is given for page rank.
I hope that helps. Google's algorithm is secret, and I'm speculating on how it has worked in the past.