I just wrote a reply for this but then remembered we have some info here all in one thread
Take a look here - http://androidforums.com/international-galaxy-s3-all-things-root/443802-samsung-galaxy-s3-all-things-root-guide-last-updated-09-june-2012-12-05-pm-gmt.html
which is the information about rooting on a Galaxy S3. There is lots of helpful stuff there
I don't think Odin is mentioned: Odin is a piece of software for Windows computers that lets you update or flash parts or all of the system. It is an internal piece of software used by Samsung but it has been leaked and is now used by a lot of people. This is what you use initially to root the device, and if you are say on an older firmware version and a new one has come out, you can download the new one, connect your phone into your pc with odin and update it.