A paddle is a game controller with a round wheel and one or more fire buttons, where the wheel is typically used to control movement of the player object along one axis of the video screen.
The paddle wheel is usually mechanically coupled to a potentiometer, so as to generate an output voltage level varying with the wheel's angle relative to a fixed reference position.
The reason for the name paddles for this type of game controller is that the first game that used it, Pong, was a (somewhat crude, but pioneering) video game simulation of table tennis, whose racquets are commonly called paddles.
Game controllers can be used to govern the movement or actions of elements in a video or computer game.
For games, the keyboard typically controls movement of the character while the mouse is used to control the game camera or used for aiming.
Since the controller is the most common way of interacting with a game, it has to be ergonomically designed to feel comfortable to most of their potential userbase to avoid injuries such as the ones in the RSI group or CTS.