Analytics
Analytics in the web context is the subject of tracking website users and deriving insights or value from those data.
Local Tools
An excellent program for analytics is AWStats. The nice thing about AWStats, aside from it being free open-source software, is that it's locally installed on your system and so there is no third party. You have your stats, and nobody else does.
wp:Webtrends is a commercial product and is focussed on SaaS model, but it supposedly is still available as a software product.
Hosted Services
Most Analytics systems today are offered as a service. Chief among those is Google Analytics (or GA for short).
In-wiki Analytics
To implement Google Analytics in your wiki, you should not use the googleAnalytics extension or others as they have some shortcomings pointed out by Dantman. Instead, he offers up a better approach: putting the code right in your LocalSettings file.
Analyzing your competitors
A few sites provide insights to any public website.