PageRank is one of Google's webmaster tools to measure the link popularity of a website. Basically put, websites with many backlinks get a higher rating than websites with less backlinks.
Search engines use the PageRank to decide where a website will be displayed in the search index.
There are other criteria for the PageRank besides the number of backlinks (e.g. the domain age, which is why older, reputed domains shouldn't be dumped for new ones if it can be avoided).