What are nofollow links, and why use them?

A Wikipedia author has defined nofollow as being nofollow is an HTML attribute value used to instruct some search engines that a hyperlink should not influence the link target's ranking in the search engine's index. It is intended to reduce the effectiveness of certain types of search engine spam, thereby improving the quality of search engine results and preventing spamdexing from occurring.
Source: http://en.wikipedia.org/wiki/Nofollow.

History of Nofollow and comment spamming

Historically, one large factor in determining how well a website would rank in search engines would be how many incoming links a web page had. The more links to a page, the more popular it was deemed to be.
This quickly led to canny webmasters and SEOs using various methods to fire as many links as possible into their websites. They would do this in many ways some completely legitimate, such as reciprocal linking, directory and CSS gallery submission, link baiting, etc.
However, another way to get links into websites very quickly was to put them there yourself. Hence, comment spamming. All webmasters had to do was find a blog, and make a comment the comment usually linked back to the website of their choice. Content spamming wasnt good for blogs many of them became swamped with useless comments the purpose of which was just to provide links back to various websites.
In 2005, Google developed the link attribute nofollow. Links with this attribute set do not influence their target sites ranking in search engines.

Blogs adding nofollow as default to outgoing links

Many blog platforms set all outgoing links to nofollowed making it much less attractive for SEOs to collect links from blogs this way.
From now on, when Google sees the attribute (rel="nofollow") on hyperlinks, those links won't get any credit when we rank websites in our search results. This isn't a negative vote for the site where the comment was posted; it's just a way to make sure that spammers get no benefit from abusing public areas like blog comments, trackbacks, and referrer lists.
Source: http://googleblog.blogspot.com/2005/01/preventing-comment-spam.html.

When should we use nofollow on our own sites?

A) The most obvious use of nofollow would be to add it as a default link attribute to any areas where the general public could add links if you have articles on your site and a comment area, it would be wise to add it to any comments.

B) If you sell banner space, or sell links from your website you may want to consider nofollowing these links. Interestingly, selling links is OK in Googles opinion, as long as they are nofollowed.

C) If you want to PR sculpt or focus your pagerank to certain pages on a site, nofollow can be used on select internal links on your website. For instance, if you have a page on your site with little content, or a page that shows content pulled from other pages you may wish to no follow all links to that page. Why give a useless page pagerank?