Network Address Translation, or NAT, is how a router translates the single Public IP address it receives from your internet service provider into the multiple Private IP addresses that it gives to your devices. You can have different NAT types based on how your router forwards or blocks ports. This mostly affects gaming devices by allowing them to communicate with other consoles, or blocking them. Different consoles label the NAT types differently, but they have the same outcome.
Type 1/Open/Type A: The device is directly connected to the internet. This is usually if a console is in a DMZ, or connected without any routers or firewalls. Consoles with this NAT type are generally able to connect to any online service.
Type 2/Moderate/Type B: The device is connected to the internet behind a single router or firewall with port forwarding. Consoles with this NAT type are typically able to connect to most services, but may have trouble if the connection requires a Type 1 NAT.
Type 3/Strict/Type D: The device is connected through multiple routers or firewalls, or the required ports are being blocked. Consoles with this NAT type are restricted in the services they can access, and may have trouble connecting to online games or voice chats.