A dofollow backlink means that Google will follow that link to your site, and that link will have a value, importance and will help you have better rankings in search
If you use firefox you can get a small plug-in called "nofollow". This color codes the links on a website/blog and allows you to quickly decide if leaving a comment or contacting the webmaster to trade links is worth your time and effort.
technically there is no such thing called dofollow. dofollow is a term that is used to mean that the link does not have a rel=nofollow attribute added to it. these are the type of links you should go after as google will follow these links and pass you pagerank/link juice
Actually this term comes from the fact that search engines introduced a new attribute to anchor tags known as nofollow. What it does is that it lets search engines know that you the anchors which are marked as nofollow will not receive link juice or PR. dofollow mean that an anchor tag is not marked with nofollow and thus page rank can will flow from the page with this anchor tag to the target page.
Technically it is <a href="http://site.com" rel="nofollow">link<a/> = it is nofollow link <a href="http://site.com">link<a/> = it is dofollow link If a link on your website is dofollow then search engine crawler will follow those link and it will be considered as you vouch for that site. It is good if linked site is relevant to your site but think what if that link is targeted to a spam or porn site ? In that case Search engine may not link your site. If link on your site is nofollow then search engine crawler will not consider that link..
If your linking site has do follow attribute than your site will get that linked site's pr juice and your ranking will be better in search engine.
I think this one explains it pretty well, learned a thing or two from it, I think it is easy enough to understand: inzoneinternet.com/another-do-follow-101-post/