Piwik is an open-source, self-hosted analytics program that's frequently cited as an alternative to Google Analytics for those who are uncertain about how Google might be using its customers' data. Like Google Analytics, it's a flexible, broad-overview platform that focuses on visitors: where they come from and what they do once they're on your site. It's useful for any kind of site, be it e-commerce, content-driven, an intranet system or a government site. You can install Piwik on your own server or pay a small fee to have a third party host it for you.
The drawbacks of Piwik that people mention have to do with real-time information and time tracking. Real-time viewer data is available, but it isn't very detailed. There's also no way to accurately tell how long visitors spend on your site unless they view multiple pages. If they stay on their landing page and leave from there, you might not even know they were there at all. And if you're looking for funnel analytics, then you'll be disappointed — you'll have to get that from another service.