What does NAT do on Xbox Live?

Microsoft defined three categories of NAT: open, moderate, and strict. Devices that perform strict or moderate NAT can limit the ability of gamers to find each other, participate in multiplayer sessions, or hear each other on Xbox Live.

What does each NAT type mean?

The classification system boils down to something called a port-assignment policy.

Open NAT means that either the port-assignment policy is minimal or the device has a fully compliant version of UPnP (Universal Plug and Play) enabled by default.

Moderate NAT means that the port-assignment policy is minimal, but the device is filtering addresses or ports.

Strict NAT means the port-assignment policy is aggressive.

How to make router/firewall to be open kind?

To make the firewall/router from strict to open, actually Microsoft′s devices looking for three UDP ports to be properly routed. If you can add following ports to the NAT rules on the router, it will be considered open:

UDP port 5060;
UDP port 5061;
UDP port 3074;