Google PageRank is a patented mathmatical algorithm that attempts to order web pages based on numerous variables.
It is symbolically displayed as a horizontal green bar that ranges from 0 to 10. The theory being a PR10 graded website will display higher in the search engine results, the site will be more authoriative and full of more useful information than a site graded as a PR0.
From Google themselves:
"When Google was founded, one key innovation was PageRank, a technology that determined the “importance” of a webpage by looking at what other pages link to it, as well as other data. Today we use more than 200 signals, including PageRank, to order websites, and we update these algorithms on a weekly basis."
Google Technology overview
For a more technical indepth analysis then visit Wikipedia.
PageRank - Wikipedia
So that's the theory.
But in reality, it has been largely superceded. Google PR has proven to be easily manipulated as website owners and SEO firms obsessed (and still do) with obtaining a higher PR in the belief of better search engine positions, and more traffic/conversions.
Today, we can see numerous examples of sites that have a lower publically displayed PR that outperform and have a much better search engine ranking then higher PR sites.
Google PageRank now has been almost demoted to something of a marketing tool that has spawned a culture of obsession, which most of us at some point are guilty of.
To compound issues, one of the main places people can see PR is by installing a Google Toolbar on your browser, or by installing a 3rd party app. Unfortunatley, these toolbar displays are only updated every 6 months, which paints an even more unaccuarate picture of the web landscape.
So should you worry about it?
Short answer - No.
Google Pagerank is just one of many signals that are used to grade your websites and pages, and determine your position in peoples searches. If you are at the top of Page 1 for a chosen keyword or phrase, it's unlikely someone will not open your page because you don't have a high enough Pagerank.

The chances are, you are there because it is relevant to the users search term.
If you write and create decent content on your website, abide by white hat SEO practices, such as a balanced link building campaign, then over time, your site will be automatically assigned a PageRank.
You may also find that your sites PageRank fluctuates, and may even go down. This is normal, it shows the fluidity of the web, and can be caused by such things as regular algorithm updates by Google. Without doing anything, you may find it increases months later.
If you want to contribute in a Pagerank Discussion, here are some PageRank Threads we have running:
Google Page Rank Discussions