PageRank is the name of Google’s link authority algorithm. Bing’s version is called Page Score.
Link authority algorithms help search engines compare eligible pages for a given query based on how often they are referenced in links on other pages. These algorithms are used heavily in determining which pages rank highest for a given search query.
In the early days of the web, search engines had a hard time returning the best, most authoritative pages for a given query. Often relying on signals webmasters controlled (like the now-defunct Meta Keywords tag or the density or occurrence rate of keywords on a page), it was common for webspam pages to rank where better results were available.
Google arrived in the late 90s, and with much better (and cleaner) search results, they quickly dominated the web search space and have ever since, becoming one of the biggest (and wealthiest) companies in history.
They were able to do so because of PageRank.
The original PageRank academic paper, published by Page and Brin, is still available on the Stanford website. This quote from the introduction summarizes the purpose, and value, of PageRank:
“In this paper, we take advantage of the link structure of the Web to produce a global ‘importance’ ranking of every web page. This ranking, called PageRank, helps search engines and users quickly make sense of the vast heterogeneity of the World Wide Web.”
In the original PageRank paper (link above), Page and Brin mention basing PageRank on a “random surfer” model – the idea was that PageRank would represent the likelihood that a random surfer, in other words a user who is as likely to click one link on a page as any other, would land on a given page.
The greater the likelihood, the higher the PageRank, the higher that page should be ranked in search results.
This model revolutionized search at the time, but it had a logical next step: since people don’t browse randomly, PageRank should consider what links are most likely to be clicked vs less obvious links on the page.
In 2004, Google filed a patent for what they called a “reasonable surfer” model for PageRank:
“This reasonable surfer model reflects the fact that not all of the links associated with a document are equally likely to be followed. Examples of unlikely followed links may include ‘Terms of Service’ links, banner advertisements, and links unrelated to the document.”
As above, the reasonable surfer model considers that spreading PageRank, or authority, evenly across all links on a page doesn’t mirror the actual likelihood that a given link will be clicked – links should be weighted by where they occur on the page, topical relevance and other factors.
The exact model for the reasonable surfer version of PageRank is not public, and as this technology is central to the quality and relevance of Google’s search results, it is likely something they continue to tweak over time.