Dofollow links do not technically exist. They are quite simply a hyperlink from one webpage to another, with no other attributes. The term 'dofollow' was quickly adopted by the web and SEO communities when Google developed the 'nofollow' link attribute in the early part of 2005. The 'nofollow' tag was introduced in an attempt to allow a webmaster to tell Google, Bing and other search engines that they do not wish to be associated with a link, and do not want any 'power' or 'link juice' to be attributed to it. In effect this neuters the link, meaning that back in the day when Google still showed us PageRank values for individual pages, this PageRank value would not be transferred via a link with the 'nofollow' attribute. You can see the patent in whole for the nofollow tag here: http://microformats.org/wiki/rel-nofollow
A dofollow link is a 'natural' link that is unhindered by any filtering and is free to transfer any weight, power, authority, link juice, whatever else you choose to call it, to the page it is linking to. For example, if 'Blog A' links to 'Blog B' with a dofollow link, they are saying that 'Blog B' receives their 'thumbs up' and the search engines should treat the link as a vote of confidence. On the other hand, if 'Blog A' links to 'Blog B' with a nofollow link, they are saying that they do not necessarily endorse 'Blog B' and that the search engines should follow it at their own risk.