Is it a measure of anything that a developer or even manager can look at and get meaning from? I know at one time, it was all about the 7, 8, 9, and 10 PageRank. But is it still a valid measure of anything? If so, what can you learn from a PageRank?
Note that I'm assuming that you have other measurements that you can analyze.
As to what I have experienced, the pagerank is an indicator for how many sites recently linked to your site. But it is not necessarily connected to your position on Google for example.
There were times where we increased our marketing and other sites linked to us, and the pagerank rose a bit.
I think the factors resulting in any SERP position are changing too much to put all your faith into one. Pagerank was very important, and still is to some degree but how much is a question I can't answer.
Every link you send out on a page passes some of the page's pagerank to where the link is pointing. The more links, the less pagerank passed on to each. Use rel="nofollow" in your links to focus pagerank flow in a more controlled manner.
PageRank is specific to Google and is a trademarked proprietary algorithm.
There are many variables in the formulas used by Google, but PageRank is primarily affected by the number of links pointing to the page, the number of internal links pointing to the page within the site and the number of pages in the site.
Thing you must consider is it's specific to a web page, not to a web site. So you need to optimize every pages.
Google sends Googlebot, its indexing robot, to spider your website, the bot is instructed not to crawl your site too deep unless it has a reasonable amount of PR (PageRank).