PageRank is a algorithm used by Google to measure the relative importance of web pages, and it was publicly shown in the Google toolbar and other browser addons, until Google stopped updating it. See also: How often is pagerank updated?
The PageRank that is publicly known was heavily based on links, which made it particularly vulnerable to the manipulation of website owners. Entire directories existed just with the purpose of providing free links for our websites, these would quickly loose their influence when Google started ignorering such links.
Typically pages with a higher PR, would also be positioned higher in the search results.
PageRank might still be used behind the scenes
While the public PageRank is not updated anymore, Google might still be maintaining their own PageRank behind the scenes. This is, however, not available to the public.
In 2014, Google’s John Mueller said in a webmaster video, that they were "probably not" going to update public pagerank abymore:
(20:25) Toolbar PageRank is something that we have not updated for about a year now, and we’re probably not going to be updating it going forward…