The way I understand NAT (Network Address Translation) is quite simple. It goes like this, the router sends and receives traffic for all the computers on the network by filtering and attaching identification headers on each packet so that external servers can communicate with machines on the router's network.
One of the other main tasks of a router is to be a Firewall. My understanding of this is that because you only have a single external ip by which hackers can enter you make it much easier to keep that one connection secure. Couple this with the fact that the router locks all extra ports by default making it very tight and secure. Granted you can create what is called a "pin hole" which simply forwards all traffic from the port of your choice to an internal ip of your choice. This is necessary if you will be running a webserver behind the router for example.