A spinner is a common arcade game controller. They were also used on several early console systems (like the Atari 2600), but for some reason Atari called them "Paddles".
A spinner controller is a knob that you rotate left or right to cause movement on the game's playfield. The first spinner controllers were used in games like Pong, Breakout, and Super Breakout (and their many clones). Spinner games dropped a bit in popularity in the early 1980s, but enjoyed a brief surge when Arkanoid was released (Arkanoid was probably the best spinner game ever made).
There are two basic variety of spinners, optical and analog. I am going to talk about optical spinners first, as they are the most common. I will also touch on a few unique types of spinners that are very uncommon.
Optical spinners are by far the most common type. They work on the same principle as the mouse on your computer. Turning the spinner turns a small encoder wheel which translates into movement inside the game. These are digital in nature, but they closely mimic analog operation. Games such as Arkanoid, Tempest, and Homo use optical spinners.
Analog spinners were mostly used in 1970s era ball and paddle games, such as Pong, Winner, and Breakout. These spinners work by turning a POT underneath the control panel. These provide true analog movement, but do not rotate nearly as freely as optical spinners (nor do the provide for full 360 degree rotation). These spinners are not as reliable, but are very cheap to repair when they do fail.
A few games use other kinds of spinners. Discs of Tron uses an optical spinner that can be pressed down or pulled up for an additional degree of control. A few other titles use microswitch based spinners that only have a few distinct positions. Finally, some games (such as Ikari Warriors), integrate spinners into their joysticks. These integrated spinners usually have a small range of disctinct positions (Ikari Warriors has a 12 position one), and are unique to the game that uses them.