A PageRank
PageRank results from a "ballot" among all the other pages on the World Wide Web about how important a page is. A hyperlink to a page counts as a vote of support.
How it's detirmined
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. If there are no links to a web page there is no support for that page.
Google assigns a numeric weighting from 0-10 for each webpage on the Internet; this PageRank denotes a site’s importance in the eyes of Google. The PageRank is derived from a theoretical probability value on a logarithmic scale
The theory
Page Ranking is derived from a theoretical probability value and includes probability distribution, damping factors
Manipulation
Various strategies to manipulate PageRank have been employed in an effort to improve search results rankings and monetise advertising links. Google is known to actively penalize Link farms and other schemes designed to artificially inflate PageRank.
Google penalizing sites selling paid text links. How Google identifies link farms and other PageRank manipulation tools are among Google's sectrets.