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PageRank is an algorithm developed by Larry Page and Sergey Brin (the guys that founded Google) while they were still studying at Stanford University. The algorithm was a central piece to their search engine project that in 1998 started to offer service under the name of Google.
Google assign a PageRank to every page it finds while crawling the web. It is often displayed as a number between 0 and 10 on a logarithm scale, which means that a page with a pagerank of 5 is ten times more "important" than a page with page rank 6. (Much like the Richter magnitude scale for earthquakes).
Having a high PageRank helps a page to appear higher on the results for a particular query in the Google search engine, and thus, in getting more hits and more traffic.
In Page and Brin's algorithm, a page linking to another is considered a 'vote' and it will increase the linked page PageRank. So if many sites link to site Z, it's PageRank will grow from all those links. In other words, Google will 'think' it is important and/or useful and will likely appear very high on results page.
However, not all the links or 'votes' weight the same, a link that comes from a high PageRank page weights more than those that came from small PageRank pages.
So in order to calculate the PageRank for a particular page, we need first to find all the links that point to it, and then, depending on their PageRank, calculate the given page one. The problem is, how do we know the Page Rank of those pages that link to the one we are working with? This is a loop-hole problem, since at some point, the linking chain if the web will close.
So, how does Google calculates all this?
To make a long story short, at the very beginning, it assigns a fixed value to a page (say 1). Then in the first few times the algorithm is run, this value can increase or drop a lot, only to stabilize at some point after many iterations in the final PageRank value.
So, if you want to increase your PageRank value, you need people to build links to your website, and as always, the best way is to provide them with good quality content.