The most popular one is obviously Google Analytics. If you are part of a big enterprise company, then Omniture is perhaps your best option. I have used AWStats as well, however I did not like it too much. Google Analytics FTW . Simply install the tracker code on all your pages and you should be set to go. It allows you to see navigation path summaries, entrance and exit points - the number of visitors you got on specific days, weeks, months and even compare them to past data. That is just the tip of the iceberg, you can do tons more - especially with their new in-page analytics tool which allows you to look at your page and tell you how many percentage of visitors clicked on any specific links. Try it out.
I never paid much attention before, I concentrated on the server stats but I've just signed up for Google Analytics and Quantcast which I hope will prove useful.
Besides Google Analytics, I also use something called REF Code Analytics (http://www.9thsphere.com/refcode-analytics.html) - it is a different analytics tool that allows you to calculate the effectiveness of your website design and the number of conversions you get not through your submission forms, but through phones calls that you get. These phone calls need to be generated through someone coming to your website and finding your number. It is a fairly strong tool that tracks the online activities of callers who have visited a business’ website prior to contacting that organization by phone. You can read more about it by searching for it on Google.
We use AWSTATS, Opentracker.net and Quantcast. I just started using Opentracker and really like it. It costs $40 a month, but that works for us because we have a very popular website and sell advertising. Opentracker tells you where your traffic comes from. So if someone accesses from a business server you know that a user has accessed from that business, although you don't know who. This is very valuable information when trying to sell advertising. AWSTATS is great because of its visual appeal and the fact that it counts everything, but segregates out robot traffic. Unfortunately, it seems rather unstable, particularly lately. Quantcast is great and also free, like AWSTATS, but there is no support and sometimes features stop working.
My friend is confused as well on the results of his blog. He said that his blogger tool reports lower traffic when compared with Google Analytics.