NAT, Network Address Translation, takes care of Internet requests from the LAN (Local Area Network) to the WAN (Wide Area Network, or in most cases the Internet).
Your home computer is at the LAN side of your router, if you try to open a webpage, NAT in your router receives the web page request and forward it to the WAN side, where your Internet modem is connected.
The reply messages of the requested website enter the router at the WAN side and NAT directs the webpage info to the computer that asked for the page.
In other words, NAT only works on commands from the LAN side of the router.
NAT takes care that the webpage you requested does not end up on the computer of your wife or child or anyone else at home.
Unsolicited messaged from the WAN side are discarded by NAT because no one at the LAN side asked for it.
If you disable NAT, you disable the WAN to LAN router function and your "router" becomes a networks switch and wireless access point only.
Port forwarding in the router allows you to direct unsolicited message from the WAN side to a certain computer on the LAN side.