What is PageRank?
First, let's clarify that PageRank is not the same as your site's ranking on search engine results pages (SERPs). PageRank is an algorithm used to determine the importance of webpages based on the quantity and quality of inbound links. It was first created by Larry Page and Sergey Brin while studying at Stanford University and would soon become an important factor used by Google to provide relevant search results.
Although Google has almost certainly made modifications since then, here is a simplified version of the original equation published by Page and Brin:
PR(A) = (1-d) + d(PR(B)/L(B) + PR(C)/L(C) + PR(D)/L(D) + ... + (PR(n)/L(n))
Where PR(A) is the PageRank of Site A, L(B) is the number of outbound links from Site B and d is a damping factor of 0.85. To put it in words, the PageRank of Site A is the sum of the PageRank of all inbound linking pages divided by the number of outbound links on those pages. Or, to put it in terms that matter for your site, more high quality inbound links means a higher PageRank for your site.
The equation can best be understood through the metaphor of a random web surfer. The surfer starts on a random webpage and clicks on a random outbound link from that page, then clicks on another random link from the next page, repeating this task indefinitely. The equation essentially calculates the probability that a surfer who is randomly clicking on links will eventually end up on a certain page. So if PR(A) = 0.25, there is a 25% chance that the surfer will randomly end up on that page. The damping factor (originally proposed to be set at 0.85 by Page and Brin) means that our surfer has a 15% chance of jumping to a random page instead of following a link. Without a damping factor, the surfer would end up being stuck in a select network of important pages (probably Google, Facebook, YouTube, Yahoo, Wikipedia, Twitter, etc.).
When considered within the immense scope of the entire internet, the actual PageRank probabilities are likely to be very small, so Google normalizes those numbers (while considering many other factors) giving each page a score on a scale of 0-10.
PageRank is just one of the 200+ factors used by Google to determine search engine results pages. There is a lot that goes into getting your site to appear on Google's first page of results, but high quality links and thus, a higher PageRank is a very important step in the right direction.
If you're interested in learning more about PageRank, Wikipedia provides more than enough information.