Dofollow and Nofollow are the link attributes attached to a link on any website to assist the search engine about its quality.
The dofollow link on a website means that it thinks that a link is credible enough to be listed on their website and they would recommend other users and search engine robots to follow this link. This is great for the link which is followed because this would increase its traffic and page rank in Google, on the other hand no harm would be done to the one who mentions this link.
A nofollow tag has the following format (HTML code): <a href=”[url]http://www.website.com”[/url] rel=”nofollow”>text link</a>.
The nofollow tag indicates to Google (and other) spiders to not follow the link to its intended place. Thus, making such links value-less from an SEO point of view. The only intended audience for these links are human visitors who cannot tell the difference between a followed link and a nofollowed link (because the code is hidden in the back-end).