The Economist has a very concise explanation of how Google works, and how it became today’s dominant search engine.
Mr Brin’s and Mr Page’s accomplishment was to devise a way to sort the results by determining which pages were likely to be most relevant. They did so using a mathematical recipe, or algorithm, called PageRank. This algorithm is at the heart of Google’s success, distinguishing it from all previous search engines and accounting for its apparently magical ability to find the most useful web pages.
Most importantly, the article gives a detailed, but easy to understand explanation of how they do this.