For the most part you don't need anything special to play the vast majority of games. You just need the Wii console. When you buy the Wii it comes with one Wii remote, the Wii’s primary controller, which looks something like a television remote, and a nunchuk, a device that attaches to the remote via a cord and is held in the opposite hand. The nunchuk is not used for all games, but it is used for the majority. The Wii remote is always necessary.
Many games have a multiplayer mode that allows up to four players to play simultaneously, so it’s a good idea to have at least one more remote and nunchuk so a friend can play along. You can either buy Nintendo’s remotes and nunchuks or try third party controllers from companies like Nyko (who for some reason calls their remote The Wand). Third party remotes are often less expensive, although some people complain they are not as reliable (I’ve been happy with Nyko, but haven’t tried other third party controllers).
Some games require more than just a remote and nunchuk. The most common necessary accessories are the MotionPlus and the Balance Board. This should be indicated on the front of the game box by a picture of the necessary peripheral encircled in the text telling you it is required. For example, Wii Fit Plus shows a picture of the Balance Board with a circle around it saying "requires Wii Balance Board" and text next to it saying the same thing. MotionPlus games can be played with either the MotionPlus add-on attached to a a Wii remote or with the MotionPlus-capable Wii Remote Plus.
Other games will just say they're "compatible" with a peripheral. For example, Punch-Out! says "compatible with Wii Balance Board," which means the game is designed so you can play it without the balance board but if you have the board it will allow you to do extra things. Sometimes those extra things are very important, sometimes they're not. Check reviews of games to gauge how important these compatible peripherals are.
It’s worth mentioning that you can’t just buy a Balance Board by itself, but only packed with Wii Sports or Wii Sports Plus.
Some peripherals are always optional, such as the Wii Wheel and the Wii Zapper.
The Wii Wheel is a wheel shell, which means it is simply designed to encase a remote. Most racing games will have a notice on the front stating that they are compatible with the Wii Wheel. Wheels are never necessary to play a game, and they don't add much in the way of functionality, but they do add to the feeling that you are driving, so many people prefer to use them.
The Wii Zapper is similar, a gun peripheral that holds a remote and nunchuk and is used in many shooting games like Call of Duty: World at War or Resident Evil: The Darkside Chronicles. Once again, you don't need it, but people feel holding a remote shaped like a gun adds to the experience. Personally I don't care for the Wii Zapper; at the moment I play shooting games using the Penguin United Crossfire Remote Pistol.
There are also add-ons for many sports games and minigames. Sets like the Nerf Sports Pack or Penguin United’s MotionPlus-compatible Active Motion Bundle for Wii Sports Resort come with attachments to turn your remote into a golf club or a ping pong paddle. You never need them, but they can be kind of fun.
Some games have their own controllers. These include the Rock Band, Guitar Hero and Dance Dance Revolution series. Often these games are bundled with their controllers, but not always. One would expect such games to state on the box what controllers they need, so I was surprised to see no such indicator on the cover of Lego Rock Band. Once again, it’s a good idea to read reviews of games before you buy them. You can also try asking a salesperson in a games store, although there is no guarantee they will know what they’re talking about, but most of them should know something as important as whether a game needs a guitar controller.
The vast majority of Wii games require only a Wii remote and nunchuk. The vast majority of games that require more than that will say so on the front of the game box. There are a lot of extras you can buy for the Wii, but if you buy the right games, you don't need to buy anything else