rel="nofollow" attribute-value pair (AVP) is an elemental microformat that can be used within a HTML hyperlink element (e.i.
rel="nofollow" is added to a link (and assuming bots are respecting the annotation) crawlers won't follow this particular link, in general. Therefore, there is no weight, equity, PageRank, etc. being flown across this link. For this reason, the tag should be used on some external links and especially paid links.
As Google's Matt Cutts has consistently recommended (see video),
rel="nofollow" should not be added to internal links in general.
In only a few instances such as a login page or shopping cart page (basically any page that has not value in terms of content), it makes sense for the internal links pointing to those pages to have a
rel="nofollow" tag. This is called "crawl prioritization". However, not only that it doesn't hurt to have a login page or a "terms and conditions" page being crawled and showing up in the index, but as Google clearly states, focusing on crawl prioritization via nofollowed links can be resource intensive and far less productive than focusing on other aspects of the site such as an intuitive navigation.
rel="nofollow" is an optional attribute-value pair of a HTML
<a href="http://www.example.com/foo/bar.html" rel="nofollow">
In the past, some webmasters were using the rel="nofollow" tag on certain links in order to increase the PageRank of others pages. Since 2009 Googlebot treats nofollowed links differently in that sense that they don't maximize the equity being flown across other links but simply block any equity from being passed through the nofollowed links.
Since search engines (that respect the annotation) do not follow those links, it increases the likelihood of other links to important pages to be seen, crawled and for those pages to be indexed.
In situations where paid links are used to link to a site it is vital to ensure rel="nofollow" tags are implemented on these links. When detected by search engines, this unnatural linking not only won't increase ranking and SERP presence but can also lead to an algorithmic or manual penalty for the site being linked to.
Robots tags are directives that allow webmasters to control how search engines crawl and index the content of a website. They are page-level signals and can be implemented in a meta tag or in an HTTP response header.