PuTTY is a client program for the SSH, Telnet and Rlogin network protocols.
These protocols are all used to run a remote session on a computer, over a network. PuTTY implements the client end of that session: the end at which the session is displayed, rather than the end at which it runs.
In really simple terms: you run PuTTY on a Windows machine, and tell it to connect to (for example) a Unix machine. PuTTY opens a window. Then, anything you type into that window is sent straight to the Unix machine, and everything the Unix machine sends back is displayed in the window. So you can work on the Unix machine as if you were sitting at its console, while actually sitting somewhere else.