PageRank, abbreviated PR, is a trademark of Google's and is irrelevant to all other search engines. PageRank is one of Google's many methods of indicating the popularity of a website as a number ranging from 0 (least popular) to 10 (most popular).
Previously I explained that links from Authority websites are more valuable than links from small websites. PageRank is a numerical representation of how valuable all those other backlinks are to your site.
PageRank, Authority websites and Link Popularity are all related. The larger your website becomes the more of a chance it will become an Authority. As it gains Authority status more and more people will link to it, thus providing votes for your website.
Each link, or vote, will help to increase your PageRank.
On a regular basis Google will count up all the votes for you and calculate a number from 0 through 10. It might require 1000 votes before you reach a PR4, but only 100 to reach PR3.
Calculating the PageRank in advance of searches allows Google to give you answers in milliseconds rather than seconds. For any given topic, Google will usually show you the highest PageRanked website in the organic results.
Now let's review an example.
Every jewelry store would like to have more engagement ring business. So let's analyze why it's so difficult to compete in Google for the keyword phrase "engagement rings."
Search results for "engagement rings" returns 14,000,000 results in 0.19 seconds.
As of this writing (January 18, 2010) the results were:
Each of these 3 websites has a PR4. Obviously there aren't any engagement ring websites with a PR5 because they would show up in the #1 spot.
But PageRank isn't everything. Since James Allen, Tacori and Kay have equal PR, Google is forced to use all their other search engine optimization criteria to determine the top order.
At this point you might be wondering how many links it takes to climb from PR1 to PR2 to PR5. No one really knows because Google keeps it a secret. Estimates claim that it takes an exponential increase by a factor of 8 to rise from one PR level to the next.
These numbers start small. PR1 is only 8 links to your website; but they grow very fast. Achieving a PR5 requires more than 8,000 links to your website.
My suggestion is to not worry about your PageRank. Take care of your website, concentrate on growing your business and the PR will accumulate over time.