The best information about this topic can be found on seomoz.org/learn-seo/mozrank I am sure all your queries will be answered.
As per CEO of SEOMOZ RAND we use the same algoritham for mozrank as that of Pagerank a little refined one.The two differences that he have added.MOZrank updated like every one month.Pagerank can change from time frame ranging to one month to 11 month.Mozrank is based on all how many good sites are linking to you.They do not have manual team to check in like Google.Google penalize pages with link selling from PR 6 to PR 3 via manual teams and other checks.Mozrank is all based on software stats.
To keep things simple, MozRank gives you a score that's based on balance of Quality + Quality Links. They measure this by looking at the authority of each website linking to your site. This is in your best interest because when getting backlinks, you also want to know that they are coming from high quality websites. The score measures from 1-10.... 10 being excellent
there is no direct relationship between mozrank and pagerank. mozrank (or correctly put: "mozRank", or "mR") is a system designed by the nice people over at seoMoz that is meant to 'mimic' pagerank effectively, but providing SEOs with a more accurate interpretation - for instance 5.28 instead of just "5". Its handy for finding pages that have been PR penalized, etc.
Moz Rank and Page Rank are similar. Moz Rank is a weighted link popularity score as is Page Rank. They differ because the start of with different weighting being assigned to a different "seed set" of trusted sites. They also have different indexed so propagation of link weight through their respective indices is different.