PageRank is an algorithm used by Google Search to rank websites in their search engine results. PageRank was named after Larry Page, one of the two founders of Google. PageRank is a way of measuring the importance of website pages. According to Google:
PageRank works by counting the number and quality of links to a page to determine a rough estimate of how important the website is. The science behind what Google has with PageRank is based off of citation theory, which was developed long before the Internet. The underlying assumption is that more important websites are likely to receive more links (votes) from other websites. Not only is it the total number of backlinks that count, it is how important those links are (quality) and Google can give 8,000+ times more credit for a great backlink. Furthermore if you have a lot of "crappy" low quality backlinks Google will penalize your web site. So when people use the term "quality of backlinks" you should understand this is of paramount importance.
Even today "PageRank" is the most important ranking factor for Google, and similar algorithms are implemented by Bing and Yahoo. The science behind the concept of PageRank is tremendously powerful. Meaning it is by far the best and most useful algorithm in search. It has been enhanced over the last 14 years, however the core scientific and mathematical concept behind PageRank is still intact.
I believe Google likes all the fake SEO experts that try to claim PageRank is not used or is not important any longer.
That is totally incorrect.
You should learn that if you separate search engine optimization into "onsite optimization" and "off site optimization" there is a fixed finite maximum of points you can receive for good onsite optimization, e.g. the title tags, content, keyword density, etc.
However with off site optimization (link building) you can earn an unlimited infinite number of ranking points from Google. This is why off site optimization is more important and it can even overpower poor onsite optimization.
I will show you a graphical representation of the mathematics behind all of this which verifies what I am explaining above. If you really want to understand all of this then you need to learn about citation theory.
Also you should study the work of Gerard Salton who was one of the greatest experts in the field of Information Retrieval and was the main Professor that taught Amit Singhal the superstar and SVP of Google Search.
A PageRank results from a mathematical algorithm based on the web graph, created by all World Wide Web pages as nodes and hyperlinks as edges, taking into consideration authority websites such as cnn.com or whitehouse.gov. The rank value indicates an importance of a particular page. A hyperlink to a page counts as a vote of support. The PageRank of a page is defined recursively and depends on the number and PageRank metric of all pages that link to it ("incoming links"). A page that is linked to by many pages with high PageRank receives a high rank itself.
Numerous academic papers concerning PageRank have been published since Page and Brin's original paper.