Page Rank is Google's method of showing whether or not a website is an authority or not. Google uses a numbering system of one through ten, with ten being the highest number. Typically companies like Google, Microsoft, or Yahoo, will have a "ten" rating. Most websites have closer to a two to five rating. Page Rank lays heavy importance on "backlinking" or links back to the web page. The more links to the page, the more likely it will get a higher Page Rank.
Trust Rank, unlike Page Rank, looks a lot closer at a website. Many webpages out there are spam, filled with lots of meaningless words. What Trust Rank does is sort the wheat from the chaff: a low Trust Rank means that a website won't appear high up I search results. In short what Google does is chooses an expert to review certain "seed" websites. Once those websites are reviewed and analyzed, then Google uses that website to choose other websites related to that page that have valid, trusted information. The more relevant the content, the more trusted it is, and the higher the Trust Rank. The philosophy is that trusted webpages that link to each other beget webpages with higher Trust Ranks.