Johansen,
NAT stands for Network Address Translation. The simplest way I can explain it (And I barely understand it) is that it is how your router is moving around received information from the world wide web to your local network. Typically NAT issues are due to your ISP's modem/gateway doing NAT (because it includes a router), and then you also have a router behind that where NAT is also active. This is referred to as "Double NAT" and causes the issues you are seeing.
You can fix this one of two ways:
Disable NAT on your router (Not recommended)
This can cause issues so unless you have another device doing NAT, I wouldnt do this.
DMZ the device you are having issues with (also not recommended)
If you are trying to minecraft from an Xbox/Xbone/XboneS as well as get on XBlive, then DMZ'ing your xbox isnt a big deal. Basically this puts that device onto the wide open internet but since you shouldnt have anything of real value on the xbox it's not a big deal.
To try to figure this out in more depth, do the following and report back.
Follow the cable from your ISP (Phone line, Coaxial line, Fiber line, etc) to the first box, give us that make and model (Example: Motorola SB5161).
Follow the line to the next box, get us the same info
Continue until you get to your computer/the end point of the network.
From there we can start to map out your network and work on a permanent fix to your issue.