For those that want anonymity is better to turn off computer and do something else.
See information displayed in Google gadget GeoIP on the right side.
That much reveals your IP address, but, beside cookies that everyone knows about, there is actually more information that leaves computer. For instance, HTTP headers that bowser exchange on every contact with any web server. It is not much, but a lot more than people suspect.
Digest answer on question what are the http headers you can see on List of HTTP headers. To see what information comes from server you can use, for instance, DJ Delorie web site HTTP Header Viewer , or to see what your browser is telling to server you can use another HTTP Header Viewer created by Eric Giguere, and finally, if you want to know both at once, try Rex Swain's HTTP Viewer.
You can see all that and much more using program Wireshark. It is serious network analyzer, not only HTTP header viewer. It has it's own training program. Although, if your only need is to see what your computer is talking to Internet reading frequently asked questions should suffice.
It is standard part of Linux, in other words, it is either on installation DVD, or it can be downloaded and installed using distribution specific installation program . For my favorite distribution openSUSE that is Software Management module that can be found in YaST (Yet another Setup Tool) .
There is much more about privacy and anonymity on the web.
Nice texts and tests on privacy and security you can find on Steve Gibson's http://www.grc.com .
There is also SecurityMetrics portscan that can tell you few more bits.