Webtrax is a free Web analytics tool that is very customizable, but not as well programmed as it could be. The author admits that there are some issues, and it doesn't appear to be under active support at this time. But it does support a number of reports and proides good information from your log files.
Dailystats is a free Web analysis program that is not intended to be your complete analytics package. Instead, Dailystats wants to give you a small sub-set of statistics that are useful for reviewing on a regular basis - such as daily. It provides information on entry pages, pageviews of each page, and referrer log analysis.
Relax is a free Web analytics tool that tells you just who is referring people to your site. It looks at search engines and search key words as well as specific referral URLs to give you precise information on who is sending customers to your site. It's not a complete analytics package, but it works well for referral information.
Piwik is an open source alternative to Google Analytics. It is very flashy with an Ajax or Web 2.0 feel to it. One of the nicest features is that you can build your own widgets to track whatever data you want to track. It runs on your PHP Web server, and requires that you have PHP PDO installed already. But if you have that it's fairly easy to install and get up and running.
StatCounter is a Web analytics tool that uses a small script that you place on each page. It can also work as a counter and display the count right on your page. The free version only counts the last 100 visitors, then it resets and starts the count over again. But within that limitation, it provides a lot of stats and reports.
The free version of SiteMeter offers a lot of good stats and reports for your site. It only provides information on the first 100 visitors, and then after that it resets and starts over. But if you need more information than that, you can upgrade to the paid version of SiteMeter. Like other non-hosted analytics tools, SiteMeter works by inserting a script on every page of your site. This gives you real-time traffic but some people are concerned about privacy implications.
MyBlogLog is a tool with many different features. The analytics are not very robust, but they are not intended to be. In fact, the goal of the MyBlogLog analytics is to provide you with information about where your visitors are going when they leave your site. This can help you to improve your site so they don't leave as quickly. I wouldn't recommend MyBlogLog as your only analytics tool, but it does a good job on the stats it provides.
If there are other free Web analytics tools that I've left out, please let me know.