Google page rank was created based on the idea of using the collective wisdom of those creating web pages to get a measure of a page’s authority. The simplest view of this idea was to count up all the links to a page and then rank pages by the number of links.
One fairly obvious weakness of that simple view is that every link counts the same. It would make more sense to count links from pages with lots of links to them as more important than other links. So that is what Google PageRank does.
The scale is logarithmic which, in the simplest view, means 2 is 10 times larger than 1 and 3 is 10 times larger than 2. The nature of pagerank also means moving from 1 to 3 is much easier than moving from 4 to 6 (or even 4 to 5).
Since it is only provided as an integer though things get a bit tricker (as 4.49 is likely reported as 4 and moving just to 4.5 is likely reported as 5). So that move wouldn’t be so hard. Meanwhile two sites with a 4 and 5 public page rank could also be site with “real” page ranks of 3.51 and 5.49. In this case they difference in their weighted popularity is nearly 100 times while in the first case it might be they are nearly identical. So the visibility even when reported by Google is not so clear.
Google initial provided these page ranks publicly and updated them about every 3 months. Google added a direction to use nofollow links for various reasons and those links are ignored for pagerank (and Google says they are ignore for both public and their own unpublished pagerank). At a certain point they started to manipulate the public “page rank” if Google’s algorithms thought something about the site might be against the dictates Google made (for example, not using the optional nofollow attribute when Google thought they should or many other things Google didn’t like).
I don’t believe they confirmed this, but it seems to me the manipulating of public page rank didn’t not necessarily result in any changes in search results. It was just a way for Google to note their displeasure. Sometimes the manipulation of the public page rank would coincide with a manual search result penalty. Google, intentionally keeps this whole area opaque.
Google continually update them internally but the public number was only updated every 3 months. In 2013 they largely stopped sharing the page rank they keep using to rank pages.
Given Google’s decision to not share a public Google Page Rank the ranks by Moz have become the new way to quickly get a glance at the authority of a page.