PageRank is from Google, Alexa Rank is from Alexa.
PageRank is based on Google's super secret algorithms.....
Alexa Rank is based more on traffic and page views.
That's about the sum of my knowledge on the two. While most of us don't find either one accurate in terms of measuring a site's quality, if your site model is based on having advertisers they do like to see those numbers be good.