PageRank is an algorithm used by the Google search engine to measure the authority of a webpage. While the details of PageRank are proprietary, it is generally believed that the number and importance of inbound links to that page are a significant factor.
PageRank was the original concept behind the creation of Google. It is based loosely on the system of citation, where a paper referenced by many other papers is considered to be more authoritative/important than a paper with few citations. Given this line of thinking, a link to a site is similar to a citation in that it implies authority.
It's key to understand that nobody really knows what goes into PageRank. Many believe that there are dozens if not hundreds of factors, but that the roots go back to the original concept of linking. It's not just volume of links either. Thousands of links by unauthoritative sites might be worth a handful of links from sites ranked as authoritative.
PageRank is often considered to be a number between 0 and 10 (with 0 being the lowest and 10 being the highest) though that is also probably incorrect. Most SEOs believe that internally the number is not an integer, but goes to a number of decimals. The belief largely comes from the Google Toolbar, which will display a page's PageRank as a number between 0 and 10. Even this is a rough approximation, as Google does not release its most up to date PageRank as a way of protecting the algorithm's details.
Finally, and perhaps most importantly, PageRank is just one of many factors that Google uses to decide where a site appears in search rankings for a particular query. It is not the only factor.