Originally it was thought that IPv4's 32-bit IP addressing system -- yielding 4,294,967,296 theoretical IP addresses -- would be adequate for all purposes. However, as the Internet grew it became apparent that something had to fill the gap between IPv4 and a future system (which would turn out to be IPv6) that would take time to develop and implement. Private IP addressing and NAT fill that gap with the private IP range.
Private IP addressing uses addresses from the class C range reserved for NAT (192.168.0.0 – 192.168.255.255). Private addresses can be assigned by the router using DHCP or be manually set, after which those addresses can communicate with one another through the router.
Private IP addresses can only be guaranteed unique to an internal network, excepting conflicts. If a directly connected computer does not have a static IP address assigned, even assigning a private IP address manually will not enable communication.
Private IP addresses cannot be directly contacted over the Internet as a computer with a public IP address can. This situation affords an extra layer of security: A network NAT device communicates with the Internet using its public IP address from an ISP and checks to see if any incoming data was requested by one of the private IP-assigned computers. If so, it is directed to that computer; if not it is typically discarded.
Another benefit of using NAT, for those who do tend to have incoming requests -- like websites, file and game servers – is the ability to quickly switch servers in the event of a crash, as the incoming traffic can all be forwarded to a back-up server very easily.