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PageRank is what Google uses to determine the importance of a web page. It's one of many factors used to determine which pages appear in search results.

The History of PageRank:
PageRank was developed by Google founders Larry Page and Sergey Brin at Stanford. At the time that Page and Brin met, search engines typically linked to pages that had the highest keyword density, which meant people could game the system by repeating the same phrase over and over to attract higher search page results.
PageRank is patented by Stanford, and the name PageRank likely comes from Larry Page.

What Does PageRank Measure?

PageRank measure's a web page's importance:
Page and Brin's theory is that the most important pages on the Internet are the pages with the most links leading to them. PageRank thinks of links as votes, where a page linking to another page is casting a vote.
This makes sense, because people do tend to link to relevant content, and pages with more links to them are usually better resources than pages that nobody links.
PageRank doesn't stop there. It also looks at the importance of the page that contains the link. Pages with higher PageRank have more weight in "voting" with their links than pages with lower PageRank. It also looks at the number of links on the page casting the "vote." Pages with more links have less weight.
This also makes a certain amount of sense. Pages that are important are probably better authorities in leading web surfers to better sources, and pages that have more links are likely to be less discriminating on where they're linking.

How Important Is PageRank?
PageRank is one of many factors that determines where your web page appears in search result ranking, but if all other factors are equal, PageRank can have significant impact on your Google rankings.