How can a site owner, get a personal information about myself while there is
a session between my computer and his computer, and What kind of
information can be gathered? 
example: I know for a fact that my IP is
visible and one can extract from it the city I live in (What else can
he/she know, and how??)

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Clarification of Question by romko-ga on 14 Mar 2006 06:43 PST

It would be great if anyone who answers the question would split it into
two categories: (1) Technical Info that could be gatherd (eg. Screen
Resolution, Connection speed, etc). (2) Private Info (Such as Country,
City etc.)

Usually, there are only a couple of ways information is gathered: your
ip address and your cookies.

Through your saved cookies, sites/companies can see what types of
websites you visit most, what sort of things you commonly search for,
and other similar information (this is how "personalized ads" are
personalized).

With your ip address, which is publicly available (unless you're using
a dynamic proxy), a remote user can get your city, state, zip code,
service provider, connection speed, OS stats, OS version, connection
time, and if they have access (authorized or unauthorized) to your
isp's databases, they can also get your full name, street address,
phone number, and other such information.

There are many other ways of getting information off of your computer,
but they are far less common.  Pretty much, you shouldn't keep any
really important personal information (credit card numbers, social
security numbers, medical/legal records, etc.) anywhere on your
computer (including browser 'auto-fill' tools or cookies, even if
they're 'secure'), as it is entirely possible for a remote user/script
to have full access to EVERYTHING on your system.  It's unlikely,
because it would require little or no security on your end and someone
who really knows what they're doing (or a highly sophisticated script)
on the other end, but you should remember that it's feasible and not
incredibly difficult...

I learned the hard way, when I start to feel paranoid hanging out at a
web site, I find a new place to hang my hat. Say adios.

Here's a page with lots of tools that will let you check for yourself
what kind of information is available from your IP address.  The
middle row of tools contains IP tests - check out the one that's third
up from the bottom: http://www.dnsstuff.com/

Thank you all who wrote a comment!