OT? Where is "address book" in my cell phone?

I had assumed that the "address book" data for my cell phone was
actually in my cell phone, in the SIMM card or some other memory area.

But when I turn on the phone and go straight to address book, it often
says to wait. Then the light goes off and I push some button and it
still says to wait, and again. If it takes so long to get it, does
that mean it's at the wireless offices? It seems to take a while
sometimes even when I have 4 or 5 bars signal strength. OTOH,
sometimes it's there immediately, I think.

I have a Motorola phone on, I think, the Cingulary/AT&T network. I
would think it's the same for all phones and all networks, right?

Thanks