NAT stands for Network Address Translation.  It is a protocol used to change the destination of information from somewhere on the internet to a computer on your local network.  Let's break that down further.  There are a myriad of internet setups out there, so I'll try to generalize the best I can while making sense to your situation.

I'm sure you have at least a cable or DSL modem in your home.  Typically this is enough to handle one computer, but often times no more than that.  So you can purchase a router.  They come with or without wireless support (Wi-Fi), but that's really irrelevant for this.  One of these devices, modem or router, effectively connects your house to the outside world of the internet.  So we've established your home network on the "inside" and the internet on the "outside."

Now, say you get on the computer and get on this website.  Your computer contacts our server with a request and neccessary information so the server knows where to send a reply.  Our server takes the request and sends a reply of data back to your computer in the form of a nifty website.  This is call "solicited traffic" because it's something you asked for - you sent a request and got a response in return.  The opposite of this would be unsolicited traffic.  For example, when you goto our website, your request to our server is unsolicited.  The server didn't ask you to contact it, in fact it's rather lazy and often happier when it has nothing to do.

The main purpose of NAT is to disallow unsolicited from reaching inside your home.  It does this by closing ports on the router and keeping them closed.  A port is like a door into your home via the internet - there are thousands of them.  The NAT is like the gatekeeper, if you will.  Most people have little need for unsolicited traffic as a lot of it can be malicious people trying to start trouble.  NAT can also be configured to allow some traffic inside your home.

You can open these ports with Demilitarized Zone (overkill and dangerous), Port Forwarding (old school, but very precise), Port Triggering (middle-aged school, less precise), and the easiest is Universal Plug and Play (the easiest way to go, also very precise).

There is an alternative to setting up NAT exceptions on your router at home.  This is called Universal Plug and Play (UPnP).  It allows an Xbox, Playstation 3, or other device to talk to the router and request the ports it needs opens.  Most routers do have UPnP as there are more and more internet appliances these days that need it.  You should be able to enable it pretty easy, just consult your router's manual if you are unsure how or try giving tech support a call.

Port Forwarding is very similar to UPnP but requires a lot more work.  First, you have to know your console's IP address on your network.  By default, all consoles request an IP address and your router assigns one to it.  The assignment is seemingly random, so you'd have to manually set an address to your console for Port Forwarding to be effective.  Then you'd have to go into your router and manually select the ports and IP address your router would then forward the unsolicited traffic to.  Very precise, but a lot of work.

Port Triggering is a little different.  Basically, when your Xbox comes on and connects to Xbox Live, your router is triggered to open the necessary ports for unsolicited traffic to goto the Xbox.  An advantage to this is it closes the ports when you turn off your console.

Lastly, a note on using Demilitarized Zone (DMZ).  This can be inheritlenty dangerous.  Basically, you pick one device (Xbox, computer, etc.) to send all unsolicited traffic to.  It can mean attacks on your Xbox that cause it to slow down, freeze up, or grant access to your network.  I highly suggest you do not use it.

UPnP is being enabled on more routers by default and your console will take advantage of it by default, it is a very simple and secure way to open ports into your home.  If you're looking for more control, first ask yourself why you really need it and then consider Port Forwarding or Port Triggering.  If all three of those options don't work, DMZ is a sure fire way to go.  Lastly, if all else fails, you can always unplug the router and plug your console directly into your modem.  Happy shooting!