PageRank is an algorithm patented by Google that measures a particular page's importance relative to other pages included in the search engine's index. It was invented in the late 1990s by Larry Page and Sergey Brin. PageRank implements the concept of link equity as a ranking factor.
PageRank approximates the likelihood that a user, randomly clicking links throughout the Internet, will arrive at that particular page. A page that is arrived at more often is likely more important and has a higher PageRank. Each page linking to antother page increases the PageRank of that other page. Pages with higher PageRank typically increase the PageRank of the other page more on that basis. you can read a few details about the PageRank algorithm at http://en.wikipedia.org/wiki/PageRank.
To view a site's PageRank, install the Google toolbar (http://toolbar.google.com) and enable the PageRank feature, or install the SearchStatus plugin for Firefox (http://www.quirk.biz/searchstatus/). One thing to note, is that the PageRank indicated by Google is a cached value, and is usually out of date.
PageRank is just one factor in the collective algorithm Google uses when building search results pages (SERPs). It is still possible that with a lower PageRank ranks above one with a higher PageRank for a particular query. PageRank is also relevance agnostic, in that it measures overall popularity using links, and not the subject shrouding them. Google currently also investigates the relevance of Links when calculating search rankings, therefor pageRank should not be the sole focus of a search engine marketer. Building relevant links will naturally contribute to a higher PageRank. Furthermore, building too many irrelevant links solely for the purpose of increasing PageRank may actually hurt the ranking of a site, because google attempts to detect and devalue irrelevant links that are presumably used to manipulate it.
PageRank is also widely by users as a trust-building factor, because users will tend to perceive sites with a high value as more reputable or authoritive. Indeed, this is what PageRank is designed to indicate. This perception is encouraged by the fact that Google penalizes spam or irrelevant sites (or individual pages) by reducing or zeroing theri PageRank.
Google PageRank isn't the only link-related ranking algorithm, but it is one of the most popular. Other algorithms include:
The Hilltop algorithm (http://www.cs.toronto.edu/georgem/hilltop/)
ExpertRank of Ask.com (http://about.ask.com/en/docs/about/ask_technology.shtml)
