Forget all server side programs – they are not accurate and you can’t do any deeper analysis on your visitors.
Google Analytics is excellent but forget it if you don’t want big G to have your data.
I would highly recommend Clicky (aff link). It’s pretty easy to use, but has some rather advanced features. For example I’ve set it up to track clicks on casinos, so I know what keywords bring relevant traffic.
You can also track campaigns (banner, PPC…) if you have any, registrations to your forum etc.
Another example: if you have forum, you can name your visitors automatically, so you know exactly what each registered visitor is doing in real time (this is illegal in some countries though).
One cool and unique feature in Clicky is tracking outbound links even if they are redirected with 301.
Web analytics is MUST HAVE if you want to make money with your website.