PageRank was invented by Google's co-founder and current CEO, Larry Page, and it's one of the determining factors in where pages rank in Google. Google views links to other websites as votes of confidence that the website has quaility content. It's not entirely democratic. Pages that have been deemed important by their higher PageRank in turn exert more influence with their links. This transfer of importance is also called"Google juice."
This is great when you're trying to make pages more important. Notice how I'm linking to other pages within About.com? It's intentional, and part of good SEO practice. It's also good practice (and just plain neighborly) to link to other relevant sites on the Web when you find them. That said, there are times when you don't want to be so charitable.
When Nofollow Works
There are occasions where you want to link to a website, but you don't want to transfer any Google juice to it. Advertising and affiliate links are a big example. These are links where you've either been paid outright to offer a link or you get paid by commission for any sales that someone else makes by following your link. If Google catches you passing PageRank from a paid link, they view it as spam, and you could end up being removed from Google's database.
Another occasion might be when you want to point out something as a bad example on the Internet. For instance, you find an example of an outright lie being told on the Internet (that never happens, right?) and you want to call attention to the misinformation but not give it any sort of Google boost.
There's an easy solution. Use the nofollow tag. Google won't follow the link, and you'll remain in good standing with the search engine. You can use a nofollow meta tag to negate links for an entire page, but this isn't necessary for every page. In fact, if you're a blogger you should be a good neighbor and give your favorite sites a boost. As long as they're not paying you for it.
You can use nofollow on individual links by simply typing rel="nofollow" after the link in the href tag. A typical link would look like:
<a href="http://thisIsJustAnExampleLink.com" rel="nofollow">Your anchor text here.</a>
That's all there is to it.
If you have a blog or forum, check through your administration settings. Chances are good that you'll be able to make all comments nofollow, and it may already be set up that way by default. That's one way to fight comment spam. You'll probably still get spam, but at least the spammers will not be rewarded with Google juice. In the olden days of the Internet, comment spam used to be a common cheap trick for boosting your site's rank.
Keep in mind that the nofollow tag doesn't remove a site from Google's database. Google doesn't follow that instance of the link, but that doesn't mean the page won't appear in the Google database from links someone else created.
Not every search engine honors nofollow links or treats them the same way. However, the majority of Web searching is done with Google, so it makes a lot of sense to stick with Google's standard on this.