Dofollow links tells the search engine to crawl a certain link which can help increase your SERP ranking.

rel=”nofollow” is a parameter added to the Anchor HTML tag that tells to search engines that the owner of the site doesn’t trust the linked page, even though he’s linking to it. Other that, or it tells to search engines that he doesn’t want to allow that the pagerank of his website to flow to the linked site.

Dofollow (as far as I know doesn’t even exist), it’s the absence of “nofollow”, thus means a regular link.
However, nofollow as far as I know doesn’t tell the search engines not to go and index that page.
Even more, some search engines might even disregard that parameter.

Dofollow link tells search engine to crawl and visit that link while nofollow does not.

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