Do follow is hyperlink inclusion that tells search engines to pass on any credibility or influence to an outbound link. When Google bot visits a website it reads website pages and includes links with do follow attribute and excludes link with no follow attribute.
Nofollow is html tag that website owners, add to links in their website for the search engines not to crawl, notice, or acknowledge a link. While a nofollow link is ignored, but popular social networking site like Twitter, Facebook, LinkedIn, Myspace sites are nofollow, but they do provide a lot of other benefits.