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 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 underlying assumption is that more important websites are likely to receive more links from other websites.
A lot goes into the calculation of a PageRank. At a bare minimum, it calculates the score based on the number of backlinks your site has, and the individual PageRank of each of those linking sites. This ultimately adds up to a weighted score though a bunch more mathematical stuff.