I thought some images may help (not meaning to be insulting, I just wasn't sure how familiar you are with the system!) :
This is the Wii remote, one of these is included in the regular system package. The remote is used for nearly all the games. I would say all, but I don't know all the games, and I've never heard of one that didn't use it at least somehow, but to be on the safe side I'll say nearly all games use the remote.
This is the nunchuk, one of these is included in the regular system package. Some games use this, some do not. For example, in the game Wii Sports (which is included in the regular system package), there are certain sports like boxing that require both a remote and a nunchuk, and certain sports that just need the remote, like bowling. A nunchuk is attached to the bottom of the remote, so there is not a way to use a nunchuk on its own. For most games, a nunchuk simply makes it easier to move around, since it has a joystick like most video game controllers.
This is Wii MotionPlus. It's what you see being attached in the image on the right, and attached in the image on the left. This was developed by Nintendo to increase the sensitivity of the Wii remote to things such as force and subtle wrist movements. It was included in the game Wii Sports Resort (available with 1 MotionPlus included, or also with 2 MotionPlus included) or it is also available separately. THIS ACCESSORY IS NOT REQUIRED UNLESS IT IS SPECIFICALLY LISTED ON THE BOX! Currently only a few games require this accessory, one of them being Wii Sports Resort. The other games that require it are all sports games. However, this addition has proved to be very well reviewed and more games being planned in the future absolutely WILL require this additional accessory. The nunchuk is able to be attached to the bottom of the MotionPlus the same way it is attached to the remote.
This is the Wii Classic Controller. Like the nunchuk, this controller is plugged into the bottom of the Wii remote, and this is how it receives its power. This Classic Controller is NOT required for ANY games currently available in stores. It was developed so that people could easily play the classic video games available for purchase on the Virtual Console. Unless you are intending to purchase games from the Virtual Console, there is no reason to own this Classic Controller.
This is the Wii Balance Board. It comes bundled with either Wii Fit or the newly released Wii Fit Plus. It is also a control option on some other games, such as skiing, snowboarding, and cheerleading. If it is an option for a game, it will be listed on the back of the game package.
This is the Wii Steering Wheel. It's just a simple plastic shell shaped like a steering wheel that has a place for you to snap in the Wii remote. One of these wheels can come bundled with the game Mario Kart Wii. Wheels are available separately. There are also different styles and colors of wheels that have been released by other video game accessory companies besides Nintendo. The steering wheel makes driving games like Mario Kart Wii easier to play (in my own opinion) but it is not in any way required by a game. It's just a nice visual aid and makes the experience a little more immersive.
There are several other accessories that work similar to the Wii Steering Wheel in that they're a shell which you snap a remote into. There are some that are guns, there are some that are fishing rods, there are some that are light sabers, there are some that are tennis rackets, there are some that are pom poms! Anything that is like this (a shell) is never required by a game, they're just fun to have around to make the experience that much more realistic.
Anything that is necessary to play a game will be listed on the back of a package with symbols. These same symbols will tell you how many people can be playing the game.
As far as number of remotes, I can't see many people needing more than 2 remotes (and 2 nunchuks, and 2 MotionPlus). Most games that are multiplayer do not have 4 people playing simultaneously. Anyone who knows differently please comment, because the only games I have that even offer 4 player are Wii Sports and Wii Sports Resort. Most games will have people take turns and you can just pass the remote from one player to the next. As far as what bundles to look for, and what should be included, I'd say the best bundles include an extra controller set (remote + nunchuk, or even remote+nunchuk+MotionPlus) because those accessories get pricey when just purchasing them singly without being bundled into a game. Best advice I can give about games that are included in Wii bundles is that reviews are your friend. Video game reviews are numerous and easy to find online (or ask people on here or other forums). Look up a game and see what people are saying about it and then decide if you think it's something your family will enjoy.
This is an incredibly long post and I hope it was useful to you, if you have any further questions, feel free to ask (I'm a bit of a Wii/Nintendo/Video Game junkie in that I do lots of research to determine what is the best and most cost-effective way to purchase).
