Google offers its own free Analytics tool, which is a great way to start collecting data on your website. The EU cookie law (requiring user consent before collecting this data) may affect its usage in the future, but for the present you seem to have the green light to carry on collecting non-personal user data in order to improve the user experience.
Google Analytics allows you to set conversion goals such as “how many people clicked through from the front page to the contact page?” and allows you to track people’s paths through your site, from page to page. Your most visited pages can be optimised to funnel new sales, and demographics data can be used to tailor content to various nationalities and locations.
Google’s free offer is indeed comprehensive, but if you want to get deep into the figures, and with live data, you can use the paid-for tools mentioned earlier, Kissmetrics or Mixpanel, both often cited as standards in site analytics and metrics. And as mentioned earlier, Piwik is free, open source and self-hosted.