These services can give you valuable information about what pages on your site are more popular, who is looking at them, where they came from, what search words they used in the search engine to find your site and where they went. Several web hosting companies provide some of these services, but here are some other ways to gather the information if your host doesn't give you access to this information or if it isn't comprehensive enough.

Stat Counter
Another free service. I have not used them, but this looks like the service to use. The tracking code is invisible on your site and they track a breakdown of your current unique, returning and repeat visitors over the lifetime of your website with browser, resolution, operating system, country, search engine, keyword, referring link, drill down and magnify visitor analysis and reporting. The only thing is I do not know how long they have been operating, but If I had a new site, I would definitely give them a try.

Web Stat
An Excellent free service with good reporting and nice graphical interfaces.
 

GoldStats
Gold Stats is a free service that displays a small icon on your web page if you use their free service (see this web site for an example ince it is suing this service). You place the icon on each page you want to track and it will give you statistics for those pages. It will give you referring sites, search engine key words, browser information, operating system and display settings information and paths visitors take through your site. Goldstats has been around for years.

Site Meter
Site meter is a free web site statistic service that gives you all of the stats you need. They use a small box or counter on each page you want to track. Site Meter has been around for years.

AWStats
A Free log file analyzer. I have not used this program so I don't have much more insight that the web site information.

MRUNIX
Another free log analyzer. I think I used this company a while ago and the product is pretty good, but If I rember correectly, I had issues with the extended log file format for an IIS (Microsoft) web server log file.

WebTrends
If you have access to your raw log file (usually a text document provided by your hosting service), then you've got to try this free trial version of Webtrends. I use this software for a corporate site I maintain and you would not believe the amount of information you can gather and the nice reports that you can run and print.

There are many more free log analyzers and site stat services, but I haven't used them much so I don't have much to say about them. The links above give you a good list, but just do a Google search for: Free Web Site Statistics, Free Log Analyzer