A De-Militarized Zone or "DMZ" machine - for the purposes of a home network - is one computer that resides at a static IP address on your LAN that has been designated to be fully exposed to the Internet. Normally, if the firewall receives a packet that does not map to a request made by one of the devices on the LAN, it just drops the packet. However, if a DMZ host has been designated, the firewall will instead route the packet to the DMZ server. That means the DMZ machine is potentially bombarded with requests from the Internet that it may or may not be prepared to handle. Many of those requests will be attempts to hack or infect the DMZ machine.
The security risk to the DMZ machine is high, so why would somebody wish to designate a DMZ machine? A good reason for having a DMZ machine is when you want a machine to be outside of the firewall in order to be a dedicated server (web server, mail server, Unreal Tournament 2004, etc.) and you are willing to isolate that machine from the rest of the machines on the LAN as much as possible. Unfortunately DMZ servers are usually used for a "bad" reason. Sometimes, it is very complicated to set up all the port forwarding firewall rules necessary for a particular game or other server. Some require three, four or more rules to be added. Some routers may not support the number and types of firewall rules that need to be added. Just designating the machine as a DMZ server seems like a good alternative because the work of adding those rules is avoided.
When you designate a computer as a DMZ machine, you've effectively placed it outside of the firewall protection provided by the router. Therefore, if you're going to have a DMZ machine, treat it as a leper. Assume it has been compromised the moment you attach it to the Internet. Don't store any sensitive data on it - up to and including reformatting the machine and reinstalling a fresh copy operating system before starting to use it as such. Only designate a machine as the DMZ machine if you are not going to use it for any other purpose. Install a software firewall with only the only open ports being those needed expressly by to the server and a few other services like outbound ports 80 and 443 (for http and https, respectively).
More importantly, the other machines on your LAN should be protected from the DMZ machine. Install personal firewalls on every other machine and device on the LAN and configure those firewalls to deny any traffic from the DMZ machine with the exception of the ports it's been specifically set up to use for whatever service it is offering. That way, if the DMZ machine does get compromised, the other machines on the LAN may remain safe.