Alexa provides the information on their site which comes from the people who have the Alexa tool bar installed. That is not the average surfer but usually folks who are techies in some way, webmasters, SEO people, marketers and so on.
This produces some very good info on some sites and most nearly nothing on sites that are not associated with what the average Alexa contributor is interested in.
The sites visitor activity is pretty much private information except when a site uses the free SiteStats which was already mentioned.
To gain better visibility in the searches you can start with adding more links.
Try using Yahoo and search
Links:
http://www.anysite.com.
This will produce a much better idea where a site is linking. Try to utilize some of the higher websites for links.
Some people utilize links from forums but high quality directories and quality websites usually are generally the best to use. Forum links are also beneficial but do not depend on them exclusively. Some use forum linking almost exclusivity and if the search engines decide this is a way to game the system which could cause forum links to lose value.
Add or update the content on your website and make sure you provide enough quality information on each page. Use only unique quality content.
Make sure everything is relevant to everything else and do good on page optimization.
Promote yourself in sites like Facebook and any of the other Social networking sites thank about starting a blog if you have the time.
Think outside of the box, perhaps free ads on the internet, Craig's list or small paid ads in some of the small weekly paper advertisers. These all can help if you have a product or service to sell.
If you have an informational site that you are looking to utilize affiliate marketing or Adsence you are pretty much dependent on developing a source of focused visitors - which is either generic search results or even PPC. I am not really knowledgable on PPS so can't be of any help there.
All these things usually take time to accomplish.
Sorry about the long post