Pagerank is a system of classifying websites developed by Google founders Sergey Brin and Larry Page while students at Stanford University. It is a key technology in their company's web search tools.
According to Google, it looks at the entire web and attempts to analyze its link structure between sites, so that it can use the wisdom of people voting for other websites by linking to them. It takes into consideration many factors, including the relative importance of the site doing the linking and the number of overall sites linking to a site.
Web Workshop has a good quote about why Pagerank is important:
"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."
Since Google places so much value on this calculation, it can affect how well you place in their search engine results for your content's keywords. While it may not be the only factor, it is still a very important indicator of how you will place versus similar sites to yours.
Also, your site's rank will help people perceive the value of your site and content. If you have a relatively new site with a low rank, you may not be seen as valuable as a well established site with a high Pagerank.