PuTTY is a client program for the SSH (and older, less secure) network protocols which allow you to run a remote session on a multi-user computer (typically running a version of Linux or Unix).
Anything you type is sent to the remote computer and anything the remote machine sends back is displayed in the Putty client.
This means that you can work on your remote Linux server (on your network or on the Internet) as if you were in front of the Linux computer.
