When choosing a log file analysis program the most important part is that it actually analyzes your log files. That part might be obvious to some, but there are so many web stats programs out there that do not analyze your log files that it can be confusing to some.
Many web stats packages (including Google Analytics) relies on javascript code to execute (via a web browser) in order to register a "hit" on the site. If you analyze log files then you'll get a lot more data (like bot visits and 404 errors on your server) and it will be more accurate.
If you want straight data without a lot of fluff (sometimes fluff is useful, though), then analog (www.analog.cx) is free and works well. You can download your log files and then process them separately so there's no load on the server itself. There are other options, though, if you'd like to run it on the web server.
I also recommend ClickTracks (www.clicktracks.com), although it can be rather costly depending on your site's traffic and what you want to analyze.
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