Link popularity is simply the number of incoming links to your site. If for example I placed a link from the main sales page of this site to your site that would be a link that search engines would see as an incoming link.
Link popularity is growing more important with each passing day. In fact link popularity alone can earn you a top ranking on Google, as it is one of the most important factors in their ranking algorithm. It is possible to get good rankings without a lot of link popularity but the more competitive the keyword, the tougher it becomes especially on Google.
Link popularity will help you in the other major search engines as well though none seem to place quite the weight on link popularity as does Google.
The number of links pointing back to your site can have 2 major benefits. One is the obvious boost in the search engines but at the same time you can increase traffic from having links on other sites.
Not all links are created equal. Just having thousands of irrelevant links will not be has helpful as having a handful of high quality links in some search engines like Google. So any link is better than none at all, but getting high quality links is still by far the better choice. If you sell lawn mowers, a link from another site selling lawn mowers may carry more weight than a link from a site selling bay clothes.
Keep in mind that each search engine determines the number of links pointing to your site from it’s own database, no 2 search engines will return the same number of links. It all depends on how many of the other sites linking to you are indexed in any given search engine.