This question has two answers as the Wii U has quite an usual way of implementing backwards compatibility. You cant just insert a Wii game into the Wii U and it will play away fine, instead the Wii U needs to run an emulated version of the Wii in order to play the game.
The original Wii controllers use the same wireless technology that the Wii U uses and as a result, the system is capable of using the Wii remote and nunchuck along with any other Wii hardware controllers without any trouble. There is a sync button on the system just like before which will allow you to connect your controller to the system and play away without having to go to any trouble.
The Wii u also has direct support for the Wii controllers which will allow you to use sync them with the system and use them as a controller. Games of course do not have the motion control support so the use for these controllers is quite limited on the Wii U, but the ability is there all the same and if you want to play your Wii games you dont need to purchase anything new. All your old controllers and hardware will work without you needing to go to any effort.