[bentcale]

DoFollow and NoFollow is an attribute given to link by Google Search Engine which indidcates

NoFollow - When NoFollow attribute is given to link Google will not follow in displaying search query result.

DoFollow - When DoFollow attribute is given to link Google will display in search query result and if you don't specify NoFollow or DoFollow then implicitly it will consider DoFollow.

[Deny]

Do follow links are link which can be crawled by bots, whereas no-follows won't be crawled by bots.

[LailaCamlet]

Do follow is allow search engines to index those links and no follow is not allow search engines to index links. In SEO both links are valuable.

[shymajones]

Do-follow - links followed by some Search engines. (Links that matters)
No-follow - the opposite.

But how to recognize that a particular website is dofollow or no follow ?

[SEO]

  See the website "Source"(right click). And more about it there: http://answers.yahoo.com/question/index?qid=20091111005634AAbqIuf

[lindybert]

By default, WordPress blogs use a rel="nofollow" attribute, so when you read of Do Follow Blogs, these are blogs that have used a plugin that removes the nofollow attribute

[mapanipi]

Search it in Google.com

[margratemary]

Do follow Back links is the most powerful way to influence search engines to redirect a lot of traffic to your site. Search engine crawlers crawl the whole content on your webpage including images, links, keywords etc.

No follow links can be relevant to your post’s content and can come from the sources which have a potential to generate high referral traffic.The site is crawled but not indexed in search engines index. While it's a no follow link you cannot get the link juice.

[quantumpro]

Google crawls and follow always doffolow links. so try to get doffollow backlinks to your website.

[Kiarafernandes]

Do-follow: it is that links which followed by search engines and matter for the websites.
No-follow:it is that links which not followed by search engines and doesn't matter for the websites.
But one thing is really important in no-follow links, there are social media sites which all are giving no-follow links. Even it is important part of SEO, SMO. Because social media sites matter for the search engines. It is good source to gaining higher traffic.

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[chndvivek]

Search it in Google.com

see nofollow means the search crawlers do not crawl where as dofollow means the search bots crawl through the link

[mahishewtha]

Do-follow links will help you gain credibility in Google's eyes and No-Follow just opposite.

[johnniewalk]

In simple words: Dofollow link's are indexed by the google and it will considered as a link,
Nofollow link's are indexed by the search engine and it will not considered as link..

[SEOKavya]

links which are followed by search engine guidelines are considered as do follow links and which are not followed are considered as no follow links.