There are numerous definition and article that provide different explanations about PageRank. But in simple words, PageRank is a numerical measurement of the importance of a web page in Google search results. This numeric value varies from lowest 0 to highest 10. As the name suggests, it is a page level ranking and not a general web site level, and so each individual "Page" in a web site has its own PageRank value.
Below are two simple and understandable definitions of Google PageRank.
According to webworkshop.net, PageRank is a numeric value that represents how important a page is on the web. Google figures that when one page links to another page, it is effectively casting a vote for the other page. The more votes that are cast for a page, the more important the page must be. Also, the importance of the page that is casting the vote determines how important the vote itself is. Google calculates a page's importance from the votes cast for it. How important each vote is is taken into account when a page's PageRank is calculated. PageRank is Google's way of deciding a page's importance. It matters because it is one of the factors that determines a page's ranking in the search results. It isn't the only factor that Google uses to rank pages, but it is an important one.
According to Wikipedia.com, "PageRank is a family of algorithms for assigning numerical weightings to hyperlinked documents (or web pages) indexed by a search engine. Its properties are much discussed by search engine optimization (SEO) experts. The popular search engine Google to help determine a page’s relevance or importance uses the PageRank system. Google™ founders Larry Page and Sergey Brin developed it while at Stanford University in 1998."