In the world of Internet each and every device has its own IP address, and it is the IP address using which one can identify a device on the network.
As the name suggests - Network Address Translation is translation of the IP address. Translation is required when you have two parties who are trying to communicate with each other. In this case these two parties are a private network and the internet respectively.
NAT operates on a router which connects two networks together, and translates the private addresses in the internal network into legal addresses before the packets are forwarded to another network.