DMZ = DeMilitarized Zone, a seperate network address scheme and connections (routes) that make it a lot safer to have a server on the internet and not expose all your computers.
NAS = Network Attached Storage, a file server device used like a Windows File Server, or FTP server. I actually have several, on is backups, one is a DLNA (Movie server) and one is general data. One is 10TB Linux server also acting as a VirtualBox virtual machine server, one is a 6TB Buffalo brand stand alone with 2 x 3TB drives. Another one is an Ubuntu based PC with 2TB internal and a 4TB external.
I make my living as an IT Technology Support Specialist and I reproduce the various environments I support, so my setup is a little different than the average home user's.
Does this help any?