There are a few things that you should know when you are looking to analyze your website.
First understand if you have choosen the right keywords for your website. You can use various tools to do that.
Once you have done this then see how your website is being optimized, in the sense if you have all the right ingredients in your site that co-relates with your keyword goal. I recommend seocentro.com to do this research.
If you have completed this step, add statcounter and google analytics into your website and start monitoring the kind of traffic you are getting. If you can identify whats working, then you should analyze why it is working. Then mature to the step of whats not working and then when you know why other things are working you can make the non-working keywords to work.
If you are doing Google Adwords, you can perfectly analyse what kind of keywords are most searched and their cost. You can understand the pattern of search from a consumer stand point and implement those things into your SEO campaign.