My competitor has a pagerank score of 3, and I have a 5 - why do they rank higher than me?

PageRank is a general indicator only, and should not be relied upon as the sole indicator of how a website is ranked in the search results.  Google factor in other issues such as keyword matching within the website content, link structure of the website, matching against page titles and a host of other issues.

PageRank is calculated based on a range of factors, but mostly it's a rating based on how many links you have to your website, and the quality of those links (whether they originate from high ranking websites).  PageRank is widely believed to follow an exponential scale, with a factor of ~5 (other figures are quoted from time to time).  So a PageRank score of 6 is 5 times better than a 5, which is 5 times better than a 4 etc.   In other words, while it's not overly hard to get a 2 or 3, beyond this you need to have established a range of quality links pointing to your website from other websites.   Google also discount PageRank scores given to websites where there is a reciprocal link back to the originating website.   Google recognised this a few years ago as a trick that SEO companies were playing, and tend to only value one way inbound links these days, although this continues to be a subject of debate within the industry.