Generally DoFollow refers to links that do not have nofollow attribute. It is recognized as a plus point for back-linking and the link will be crawled by the search engine spiders. If the link is dofollow, the page rank of the webpage can be transferred into your site. No follow simply means that the webmaster does not want to have anything to do with the link. But there are occasions that we need NoFollow links as well.
In other words Nofollow links are not always ignore. There are several benefits by having no follow links as well. Having a mixture of nofollow and dofollow backlinks pointing to your site would make your link profile look more natural. Plus, not every search engine's crawlers use the nofollow attribute at all.
In DoFollow Backlinks, there's no need to put 'dofollow' attribute to the html. It would simply look like below.
<a href = "URL">ANCHOR TEXT</a>
But a 'nofollow' link has the nofollow attribution like below.
<a href = "URL rel=nofollow">ANCHOR TEXT</a>
Even-though you put your link without NoFollow attribution, if the posting blog's CSS has designed to convert it into nofollow, then your attempt to create the DoFollow link is wasted. Most of the Wordpress Blogs are NoFollow on its nature. So they are seemed to be useless in terms of backlink building.
If you are interested in buying links or hiring someone in creating backlinks, then you might want to check if a link is DoFollow or NoFollow before getting into any deal. otherwise money, time and all the resources would be wasted. Hence always try to keep track of your backlincks.