Stephen's blog post covers Google Analytics. The one major disadvantage that I see with Google Analytics is that the stats are a day behind. I want my stats, and I want them now!
I have been using StatCounter.com for my stats.
I actually got turned onto StatCounter from my fellow blogger Scott for another project.
StatCounter has a free stat offering that doesn't display any chiclet or any visible signs like other stat counters.
They do have a paid version that allows you to keep log file entries, but for most, the free counter will work just fine.
I have encountered so many business owners and webmasters who don't have access to web logs or the ability to use weblog analyzers (you would be surprised to know that some very big websites in both adult and mainstream don't have adequate access or use of tools for weblog analysis).
The use of third party stat systems that can be inserted with a simple copy and paste is easy to do and outsources the analysis to some third party whose sole focus is to deliver a great product. While there are many stat counters, I will focus on what I like and how I use statcounter.
First thing you see once you set up your free statcounter account is a summary bar that shows Total Vistors vs. Unique Visitors vs. Returning Visitors. This is a great overview to see how many people are coming to your site so you can quickly view your traffic.
The "Came From" link shows you the referring URLs, or the web pages/websites where someone had a link to your website/web page. This is invaluable to verify who is linking to you for link trades, and to spot new linking relationships. A blogger may have linked to you and by viewing this section, you can now identify the new source of traffic.
"Keyword Analysis" lets you see what keywords surfers were using in search engines that clicked on a search result to find your website. This gives you some insight into what surfers are looking for as they found your website. You can put in the exact keywords into a search engine to see how you ranked in the result. This will allow you some gauge as to where you are in a search result for that search (results may slightly vary due to the way the search engines do their querying).
My favorite link is the "Visitor Paths" because it allows me to see what the surfer was doing on the website, in moving through the sites. Combined with "Popular Pages", "Entry Pages", and "Exit Pages" you can get a sense for what surfers are looking at, and make adjustments to your website.
You may find you need to change a tour, or change your navigation, or put up some extra links/buttons to draw attention to certain parts of your website that surfers aren't finding, that you want them to find.
There is alot of tweaking that can be done to your website based on the quick information you gleem from a stat program. Having this kind of information allows you have some insight into who is visiting your website and monitorig what they are doing. Whether you are using internal weblog analysis or a third-party system, if you are not watching your weblogs and taking some time to analyze the traffic, then you are missing out on some great statistics and observations that might change the way you do things on your website, that could increase/monetize the interaction with your website offering. Fight the big brother!