Cookies are nothing more than small text files, they are sent from the server to the client and stored there. The client send them back with every page request for the use of the server.
Cookies are never stored on the server, however, if you are using PHPs sessions the variables defined _will_ use some hdd space on your server to keep the variables. This does have the plus of your scripts being a little more secure because all the client gets is a unique identifier to distinguish which text file to use.
Hope that clears things up --BB