The Classic Controller is an accessory for the Wii, which allows players to use a more old-school type of controller, with joysticks and L and R shoulder buttons. It can be plugged into the Wii Remote outlet on the bottom, and from there, players can control whatever game they are playing with the Classic Controller alone (unlike the Nunchuk which works side by side with the Remote.)
Mario Kart Wii, a Wii game that allows pretty much all controller options, lets players utilize the Classic Controller.
The A and B buttons control the Acceleration and braking in Mario Kart Wii, and the left joystick handles the steering. Due to the Classic Controller not having motion detection, pressing the D-pad lets players perform tricks... L and R deal with Items and Drifting...that's about it.
Like the Nunchuk, it was produced in a few different colors, including white (standard), black, and even golden.
Several upgrades have been made to the Classic Controller, including two lower handles for a better grip (this version is pretty popular, 'cause its even more classy, and plus, nice grips). The Wii U introduced the Classic Controller Pro, which has a similar appearance to the one with handles (some don't even have wires!), and has added upgrades.
Prices for standard and updated Classic Controllers are around $20 bucks, and Wii U version are actually cheaper, at $14.99.