Google Page Rank is a system that Google uses to rate the relevance of websites, in relation to keyword searches. Page Rank determines the order in which websites appear in a Google keyword search.
These ranked results are also called the 'Natural' search results. The 'Natural' results are those that are listed in the main middle area of the page. These are different from the paid advertising results that you will see at the very top in bold or blue, and at the right column. The paid results are provided through pay-per-click Google Adwords campaigns.
Example. Let's say I search for "Landscaping Tricks" on Google. Google will find all the sites that relate to this search, and list them in order. The bigger, more 'important' landscaping sites will show up first, in order of their Page Rank.