Not sure exactly.

If you did a Google Backlinks search (link:anydomain.com) the search results pages for some reason never show all backlinks.

Yes. Recently I had a site jump from 3 to 5 and another from 0 to 3.

I went from 0 to 4.

The quality/PR of the backlink is what is calculated, as well as average user bahavior on the page.

It is based on the QUALITY of the backlinks as well as other ONSITE SEO things. When you say QUALITY of the backlinks, you have to get backlinks from the sources which are not SPAMMY pages. Also, if the backlinks are from the sources which are not related to your niche that is fine, however, if the backlinks are related to your niche that will be great and google loves it a lot and offer your some good PR.

Google page rank is a numeric value used for describing the importance of your page on the web. Google refers to this as a method of evaluating the quality, importance and usefulness of your website. The page rank of a webpage holds a different meaning for different individuals, but it is a major sign of accomplishment for the webmaster.

The best way to determine page rank is through backlinks. Backlink is a term used to indicate a situation when some other page gets linked to your page. You get voted for each page that gets linked to your website, because of this fact it is always better to have a couple of links to your site from high page rank websites compared to having hundreds of links from low page rank websites.

Pagerank is a relative index of proximity to the "center of the web," basically Google and a few other sites. The assumption is those sites at the center of the web link to the rest of the web. This was probably a valid assumption when pagerank was introduced.

Pagerank doesn't mean much, it certainly doesn't mean high in SERPs for any given query. Primarily it means a page can give a lot of links, dispersing its pagerank, without damaging its "center-of-the-web" relativity. For example Yahoo's directory starts from PR8, and needs to, because the whole directory is obviously links to other sites, and it's sensible to consider Yahoo to be near the center of the web. Google's PR10 - in theory it links to everything, and so is a (or "the") center of the web.

Where the whole thing falls apart - sites that went on line early tend to have overly-high pagerank. Sites that are completely irrelevant today still have high pagerank (like altavista). Odds are good the pagerank disclosed in tools like the G toolbar is not the real pagerank. It's just a thing, it means more to google than it should to somebody trying to get organic traffic.

It comes down to links from high-pagerank pages and sites are good. Will they accomplish any goal in and of themselves? Probably not. They're just good.