NAT stands for Network Address Translation.
Now you have 2 devices in sequence, that translate 1 external IP address to many internal IP addresses.
NAT is used, because normally your ISP will issue you just 1 public IP address (for regular home use). Now, if you have 2 or more computers at home that would like to connect to Internet - you're in trouble : you have just one address from your ISP. And the IP addresses need to be unique across all Internet, by definition. So your 2 devices can't share this address (at least use it concurrently).
That's where NAT comes in handy - it will share that 1 IP address between all your home equipment. That's also the reason your home computers get all very different IP addresses (usually in the form of 10.0.x.x or 192.168.x.x) from the one ISP assigned you.
Obviously, all your Internet partners outside will see all your connections coming from your public (ie ISP-assigned) address, not the NAT'd inside address (like 192.168.x.x).