Pagerank is a number between 0 and 10 assigned to the web pages according to number and quality of links it gets. This algorithm is invented by Google and made Google the best search engine on the internet.
The idea behing pagerank algorithm is that: If a web page contains quality content, it will be linked by other web pages. In other words, a web page, which gets lots of external links, must be giving what people is looking for. Assume that I finished this website and uplished it. If my website contains the best build website tutorials, it will be liked by its visitors and they will suggest my website to their friends at forums, at blogs and at their websites. They will give links to my website. Thus, my website will get lots of external links and it will increase my websites pagerank. As a result, Google will see that my website is suggested by lots of people and that's why, it will increase my websites pagerank and give my website a better position at search engine results.
How to Increase Pagerank
In order to get better position at search results, your website should get a lot of links from the other websites. You can make other websites link to your website too. For example you can submit your website to web directories or send articles about your websites to some blogs etc but the most efficient way is to build a perfect website with perfect content. This will let you get links automatically.
Links and Quality
Getting links from other websites increase your websites rank but we cant say that all links have the same and equal effect. For example, link from webpage which have pagerank 8 and link from webpage which has pagerank 1 does not effect your website's rank at the same and equal rate. The link from pagerank 8 website is thousands times beneficial than pagerank 1 webpage.