Backlinks are (usually) SEO goodness for your site. These are connections from another site to yours and represent an acknowledgment of your website by the remote site. Google and other search engines examine the backlinks to your site to determine its ‘authority’.
Note that Google polices backlink quality carefully, using its Penguin spam detection sub-system to determine good from bad backlinks. If your site has ‘bad’ backlinks it may incurr Google’s wrath in the form a ranking penalty.
A nofollow tag can be added to the backlink to remove the backlink ‘SEO goodness’. This is typically done to avoid incurring Penguin penalties by contravening Google’s Quality Guidelines eg links on paid advertising sites etc Read more
This simple piece of HTML code does the work: rel=”nofollow”
Note that this variation rel=”nofollow external” also indicates that the link is nofollow and external from the site.
Matt Cutts from Google explains nofollow backlinks in detail below: