Google's PageRank has become one of the more important indicators of a website or webpage's popularity. The higher the PageRank, the better a site's chances of appearing as the top result in a Google search, or at least be in the front page for results in certain keywords. This is why the Seach Engine Optimization (SEO) business has boomed.
SEOs focus on several things to improve pagerank, and these include on-site and off-site optimizations. Google updates pageranks a few times per year, and even this is not announced. Also, Google's actual PageRank algorithm is kept secret, but there are known ways to help your site improve in ranking. Note that PageRank is a number from 0 to 10, with 0 being the lowest and 10 being highest. PR 10 and even 9 are the rarest, and these are mostly just theoretical levels, for Google's own pages ever get these.
Number of links - Traditionally, the more inbound links that point to your website, the better your chances to improve in PageRank. This is the quantity factor. And so search optimizers try to get as much links to your website as possible. This is also the reason why the "text link ad" business has thrived in the past few years.
Quality of links - Even more important than the number of inbound links, the quality of the sites linking to your website is another factor. The higher the PageRank of the site that links to your webpage, the higher the effect on how your own PageRank turns out. So a site that has a PageRank of 7 will have a bigger effect on your own PageRank than a hundred sites with a PageRank of 2.
Link neighborhood - Another important factor in improving your PageRank is link neighborhood. Aside from quality and quantity of inbound links, Google also takes a look at the quality of sites that you link to. If your website links to sites that are considered bad by Google, such as pornography, malware and phishing sites, this will have a negative impact on your PageRank. But if you link only to relevant and good-quality sites, then your chances of higher PageRank will be improved. Also, be sure not to stuff your blog with links, especially irrelevant ones. Google weighs the inbound to outbound link ratio in computing for a particular page's PageRank.
Other considerations - While an important aspect of search optimization, PageRank is not the only figure that will define whether your site appears on top or in the front page of a Google search. On-site optimization is also important, and this includes the keyword density and relevance, and the easy by which Google's search bots can index your site. Anchor texts are also important. They keywords that other sites use to link to you will be used by Google in determining the relevance of your site for those particular keywords. The speed by which your site is being linked to has an effect. If you suddenly get thousands of inbound links in a span of a week, then Google might think you are buying links en masse, and your site might be sandboxed, which means it doesn't appear in Google searches until it is verified as a valid site.
Optimizing for a better PageRank is a mix of science, art and even a bit of Internet magic. The best way to go about with it is not to force it too much, especially with inbound linkages. Focus on quality rather than quantity, because this is what will establish a good PageRank in the long term.