Pagerank which is an integer value out of 10 is applied for every page on the web which determines the value of a page from a Search Engine(Google) point of view. Alexa rank determines the approx traffic rank of a website compared to all the other sites in the world. The lesser the alexa rank, higher your traffic.
Alexa rank is assigned by alexa.com while PageRank is google technology. There are many other differences including described above.
Page rank is given by google it despond on website quality, traffic, back links. Alexa ranking despond on the traffic. Both are important to have for website.
Google have made their 10 standers 1 to 10. Page rank 5 means your site is under Google's 5th stander. While Alexa ranking (traffic ranking) means your site is on the number (Alexa ranking number) in traffic out of all websites Google recognize.
Here is a very good resource (with a video!) for you to better understand PageRank and how it works: What is PageRank and Why Does it Matter?
Page rank is algo to measure web page quality & alexa rank is same but has different criteria to measure quality of web page.
PageRank is the evaluation of the webpage in the eyes of Google. Whereas Alexa is the ranking given to websites based on the traffic that they get. However, Alexa ranking has been quite manipulative since a long time and most of the webmasters around the world have stopped worrying too much about the same.
the simple difference is Google pagerank measures popularity by backlinks and Alexa measures popularity by traffic.
alexa rank reflect traffic popularity of the site (no matter what was the way this popularity achieved) PR reflect google's trust in your site