Accessing the service: Go to the Google Analytics home page. Signing up with the service is free, except for some advanced reporting capabilities.
They offer a tour of the features and the Conversion University both geared to helping you understand what you're signing up for, and what you can get out of using the service.
Integrating the service: Webmasters use the service by inserting some javascript code into each page on their site. The javascript is expected to run on every page access, and send some information to the Google Analytics servers. That information is then crunched to produce the reports the webmaster uses.
Obviously there's a dependency here on javascript code being executed by the web browser. This means several things. For example a common recommendation to improve security is to turn off javascript, so the user who have turned off javascript will not be counted by Google Analytics. Further, spiders such as googlebot do not run javascript, hence your reports won't be showing spider accesses. That's both a good and a bad thing, as most reporting packages have to filter out spider accesses, but sometimes webmasters want to know about spider accesses.
Gathering data: