PageRank (PR) is the term that is trademarked and an algorithm patented by Google. PageRank is the link analysis algorithm named after one of the founders of Google, Larry Page. The algorithm assigns a numerical value to each page on the web based on the number of links to that page.
The algorithm was originally based on the principle that the more links there were to a page, then the more important that page must be, so therefore it should rank higher in the search results. However, with link building gone mad and the use of spam link exchanges and the creation of links for links sake with no reference to the quality of the page being linked to, Google have probably made a number of undisclosed changes to the PR calculations.
The PR of a page can be seen if you use the Google Toolbar. Google do update PR internally on a daily basis and factor that into the search results on a daily basis. They used to update PR and search rankings on a monthly basis which used to be called the Google dance, but they have not done this since 2003. The PR we now see in the toolbar is only exported by Google 3-5 times a year (see the dates of these updates or exports).
PR was also available to those who made use of Google’s Webmaster Tools and it displayed the PR distribution for a site. They recently removed in from the Tools as “We’ve been telling people for a long time that they shouldn’t focus on PageRank so much; many site owners seem to think it’s the most important metric for them to track, which is simply not true. We removed it because we felt it was silly to tell people not to think about it, but then to show them the data, implying that they should look at it.”
It is important not confuse the ranking of a page with PageRank. They are two different beasts. The ranking of a page in the search results is due to several hundred factors, of which one will be PR. The current role of PR in ranking of site not clear. Many like to be dismissive of the ‘green bar’ on the Google Toolbar. This is probably a good idea as too many obsess over it. If the links to a site play a role in the ranking of a site and if PR is a measure of the number and quality of the links to a site, then PR must have some importance (it is just the external PR value in the toolbar that probably does not have much importance). Internal PR has to have some importance as it is a measure of the links to a site.
PR is probably also important for crawling, as Google do tend to crawl deeper and index more of a site if there are more links to a site, which is important to a forum if you want all the threads indexed in search engines. PR is a measure of the links, so PR has to have some importance here.
Bing have their Page Score, but it is not known exactly what this is based on and how exactly it correlates to Google’s PR score. For more detailed information on PageRank, see Wikipedia’s article on PR; Google’s explanation; and this bit of fun.