Network Address Translation (NAT) is a technique that allows one server or router to provide one IP address publically but actually have the requests to this one address passed on to multiple servers each of which have their own IP address.
So NAT allows a single device, such as a router or server, to act as an agent between a public network such as the internet and a private network. The owners of the private network usually will not want to expose all the IP addresses of the internal servers to the public so what happens is that one address is exposed to the public and this one address is then mapped onto multiple internal servers who can handle the public requests. This in turn means that only a single unique IP address is required to represent multiple servers.
A example is where a website has one IP address but web page requests are load balanced across multiple web servers who respond to the web page requests. Each of the web servers will have their own IP address but these private addresses will not be visible to the browsers making the web page requests. The browser only sees the one public IP address. This gives the browser the impression that there is only one computer or server handling the request but in reality there can be tens or even hundreds of web servers handling the page requests.