In case you are unfamiliar with the Nunchuk, it is an accessory that simply plugs into the bottom of any Wii Remote. You may have seen Nintendo saying things like "with the Wii Remote, to hit a tennis ball, simply swing the Wii Remote". This is true for some games, for many others, it is not that simple.
How would one play a game like The Legend of Zelda: Twilight Princess without a Nunchuk? How would you control your character and swing your sword? Would you simply use the + control pad (on top of the Wii Remote) to move your character, and when you get some action, start swinging? How would you move around while in combat? Why am I asking so many questions?
This is where the the Nunchuk adds depth and variety to games like Zelda. The Nunchuk has two buttons (Z and C) and an Analog Stick. In Zelda, the Analog Stick is used for character movement, the Z button is for aiming, and the C button is used to look around. Some games use them (Zelda, Metroid Prime 3 etc.) and some don't (Boxing in Wii Sports... I think that's it). But the most innovative feature of the Nunchuk isn't ON the Nunchuck, but rather IN it.
It has a built-in accelerometer, which in human terms, means that it shares some of the motion-sensing abilities with the Wii Remote. What kind of motion-sensiing abilites you may ask? I'll give some examples.
When you shake the Nunchuk in Zelda, you do a spin-attack. When you "punch" with it in Boxing (included in Wii Sports), you punch the character you are fighting against. It may not have the pointing abilites of the Wii Remote, but it shares the core motion-sensing abilites, which it also shares with products like the iPhone and SIXAXIS PS3 controller.
To summarize, the Nunchuk attachment isn't only a cool add-on for the Wii Remote, but it is a neccessary add-on for the Wii Remote. It will set you back around $20, but it is better to buy it now than hear your kids endlessly nag you about it. They are very durable, very innovative, and very fun.