Where do I install the flash communication server? on my server or on my
computer? how do I install it on the server? I just upload some files? what do
I do with all this?
I just wanted to easily record and save video on my server...

you install flash comm server on your server.

if you have bought it (i doubt) , then it means you are going to install it on
a server for other users to see what you did.

if you want to develop then you install it on your computer.
(if you are the only developer)

if you are a company of developers then you can install it on your internal
servers as a development platform.( is not recommended since if u are many
working on it, you will get hit by its connection limit ).

Why are there so many people using flash comm illegaly ?!?!
It is not expensive to host your application on performat 3rd party servers.
It is between 20 - 80 $ for connections between 50 to 300 users, 50 users is
more than enough if you write your videochat application.