PageRank is a numeric value that represents how important a page is on the web - when one page links to another page, it is effectively casting a vote for the other page.

The more votes that are cast for a page, the more important the page must be. Also, the importance of the page that is casting the vote determines how important the vote itself is. Google calculates a page's importance from the votes cast for it. How important each vote is is taken into account when a page's PageRank is calculated.

PageRank is Google's way of deciding a page's importance. It matters because it is one of the factors that determines a page's ranking in the search results. It isn't the only factor that Google uses to rank pages, but it is an important one. 

Keywords and other non-PageRank factors can be absolutely crucial Google can put a PR6 site above a PR8 site - what this shows is that all the time people spend on exchanging and acquiring links for the sole purpose of increasing PageRank may be better spent developing website content and keyword strategies instead.

Good content tends to have the effect of both increasing your one-way in-bound links (people like to link to sites they find interesting, thoughtful, informative, or helpful), thereby improving your PageRank, while at the same time producing keyword rich webpages good for both human viewers and search engine spiders.

From Google:  Sites' positions in our [Googles] search results are determined based on a number of factors designed to provide end-users with helpful, accurate search results .... In general, web designers can improve the rank of .... sites by increasing the number of high-quality sites that link to their pages. 

Useful Resources: 

PageRank information from Wikipedia