PuTTY is a SSH, Telnet and Rlogin client.
SSH, Telnet and Rlogin are three ways of doing the same thing: logging in to a multi-user computer from another computer, over a network. Using this type of interface, there is no need for you to be sitting at the same machine you are typing commands to. The commands, and responses, can be sent over a network, so you can sit at one computer and give commands to another one, or even to more than one. SSH, Telnet and Rlogin are network protocols that allow you to do this. On the computer you sit at, you run a client, which makes a network connection to the other computer (the server). The network connection carries your keystrokes and commands from the client to the server, and carries the server's responses back to you.