This happens around launch time, it's hard to get information on what the system actually does. This video shows very well on how the Wii works: http://youtube.com/watch?v=ARQKcWJHRp0
But just to explain it, the Wii is an all-new system/console, and yes, it has connectivity with the Nintendo DS. Although, I am not so sure that it allows you to play the DS on the big screen, but you don't realy need that, because I don't think your TV has a touch screen. xD (the gamecube has no connectivity with the DS.)
Anyway, the Wii is, in my opinion, the best console out of the 3 next-generation ones. The controller, called the Wii-Mote, is motion sensitive. It senses all 3 dimensions in a vector of space.
So basically what that means is, you can do things, and your charector on screen will mimic you. For instance, if you are playing Zelda, when you swing the Wii-Mote like a sword, the charector also swings his sword. In first person shooters, you actually point and shoot, instead of just using a joystick on a controller.
The Wii comes bundled with a game called WiiSports, and that game has 5 mini games in it. They are: Baseball, Tennis, Bowling, boxing, and golf. And for each game, you swing the Wii-Mote like you are actually holding the real item you use for the sport in your hand. In bowling, you actually make a motion like you are throwing a bowling ball.
Now the Wii isn't just for hardcore gamers, it is actually desinged to try to get people who haven't played games before get into video games. (Of course, it is also great for experienced gamers.) Some of the games are very easy to learn, and you can pretty much just pick up the Wii-Mote and play.
Now you may be wondering that the Wii-Mote might not work, and it's just Nintendo saying it will. Well, I have played it before, and I can personally tell you that it works just fine. As well as video game reviewers who have spent many hours with the Wii will tell you that it works amazingly, and the Wii-Mote is very accurate.
I hope you decide to get it, it's a very good system, and your whole family will enjoy it.
EDIT: As the person below me has said, there is even MORE to the Wii, and it would actually take an hour or two to type EVERYTHING out in detail. There is even something that nobody else knows, and Nintendo will reveal it when the Wii launches on November 19th.