Is it possible to get a value of a session variable of a website from my computer which is running Apache & PHP?
yes if you give me access to your linux box.
I don't understand?! Let's take an example here: I open www.phpbuilder.com in an IE Window. Now I have a file aa.php in my computer. Is it possible to run this aa.php in the same IE window and be able to see session var in this website?
You need to run PHP page on the same computer (for example, check who is hoster of phpbuilder.com and try to get hosting on the same server) or you need to read /tmp directory on this server where session files usually stored.