"NAT" stands for Network Address Translation. A NAT functions by basically hiding the entire network behind the public IP of the computer connected to the ISP.
The computer with the Internet connection will have two IPs, one public and one private. If this computer uses a dial-up adapter to connect to the ISP, its public IP address will be in the modem's TCP/IP stack. The private IP address will be assigned to the computer's NIC TCP/IP stack. A Win95/98 computer with a DSL or cable modem connection will have two NIC's: one for the Internet connection, and the other for the private network.
NAT software maps all of the private IP addresses on the network to the public IP of the computer with the Internet connection. The software makes the Internet computer into a makeshift router. Computers on the private network will send packets to the NAT computer, and the NAT computer will replace the source address of each packet with its own public IP and then forward them to the Internet.
The NAT computer will then route the incoming packets from the Internet to the proper computer on the private network. The NAT functions transparently. The systems on the private network still maintain sessions with the computers on the outside, and most applications work fine with NATs.