What is link popularity? 
Every link the search engines find from another source is essentially a vote for, or recommendation of, that page.  More links generally means greater link popularity.  While not all links are created equal, the more links there are to a website from reputable or relevant websites, the greater the link popularity of the site.  These links are often referred to as "back links" or "inlinks".  For those of you who may have skipped over the Off-Page SEO introduction page, THERE IS ABSOLUTELY NO OTHER SINGLE ASPECT OF SEO THAT HAS A GREATER INFLUENCE ON A SITE'S RANKING THAN LINK POPULARITY. 

There are a variety of sources available to help increase link popularity.  Common sources include online directories, social media, social networking and social bookmarking sources - like MySpace, Facebook, LinkedIn, del.icio.us, Digg, Social Marker, etc.). 

Building link popularity is the most effective way to develop a high ranking website.  Building link popularity is something that can begin as soon as a site goes live.  While it is essential that On-Page SEO techniques also be implemented, those efforts should happen simultaneously with building link popularity.  The biggest advantage to building great link popularity is that not only does it help increase the ranking of the site in search engines like Google, it also exposes the site to a larger audience - potentially creating additional traffic directly from the links themselves - especially if those links appear on popular social networking sources like MySpace, Facebook and LinkedIn.  It's a win-win.

How Many External Links Should A Site Have?
Think of link popularity like "word of mouth" advertising for the information age.  From the perspective of the search engines, each external link to your site is a recommendation of the site - like "word of mouth" advertising that never shuts up.  For that reason, the more links there are to a site, the better.  The assumption by the search engines is that if a site has a lot of links to it, it must be a good source of information.  The more links there are to a site, the more the search engines trust that it is reliable and reputable - just like the more good reviews a movie or restaurant receives, the more likely people are to go.  A site with no link popularity will face difficulty being found in a keyword search - just like a restaurant would fail to generate business if no one knows it's there.

Not All External Links Are Created Equal
We've already established that search engines are smarter than they used to be.  What does that mean for link exchanges and other reciprocal link programs?  It means DON'T DO IT!  Organized link exchanges and companies that market link exchanges will often do nothing more than create reciprocal links (meaning a link from some source to your site, but also a link from your site back to that source).  If a large number of links to a site are reciprocal links, the search engines will devalue both links significantly.  Don't ask how much.  No one, except perhaps the search engines themselves, could possibly answer that. 

Participating in link exchanges and other reciprocal linking programs lowers the value of both links for a number of reasons.  First, if the majority of the links on a site are reciprocal, the search engines recognize that and they understand those links exist solely because the website owners came to an agreement to "put a link on your site and I'll put a link to your site on mine...", not because either site is considered a relevant or a worthwhile source of information.

That said, not all reciprocal links are bad.  If a site for a local plumber contains a link to the Delta Faucet and/or Home Depot websites and vice versa, those would be considered valueable reciprocal links because the sites are in relative industries.  There is something worthwhile to be gained by a visitor following either link.  On the flip side, if the same plumber had a brother who was an auto mechanic, a reciprocal link to that site would not carry as much value as far as the search engines are concerned.

PageRank
PageRank is Google's interpretation of link popularity and authority.  Many people think the "Page" in PageRank references the ranking of the website "page", but that isn't the case.  The term PageRank is actually a reference to one of the founders of the ranking system, Larry Page.  The PageRank of a site is a numerical value between 1 and 10 that is determined by the number and quality of links to that URL. 

Essentially, the greater a site's link popularity, the higher it's PageRank score.  External links from sites with a higher PageRank have more of an impact on a site's ranking in Google.  All other things being equal, if two pages are competing for the same keywords and one page has 100 external links from sites with an average Page Rank of 6 and the other site has 100 external links with an average Page Rank of 2, the first page will achieve a higher ranking because the PageRank of the sites linking to it are higher. 

It is important to note that PageRank statistics apply to individual URLs, not the website as a whole.  Every website page has it's own URL.  That said, the home page of a site might have a PageRank of 5, while another page of the same site might only have a PageRank of 1 (or vice versa).