I'm completely new in the world of video games, so forgive me if I'm clueless. I just bought a used Wii and found out that it was modified so that it can play N64 and GameCube games. I'm not sure what else it can do. I'm wondering which controllers I need to play N64 and GameCube games on it. I have the standard Wii controllers and an N64 controller that has some kind of USB adapter for playing N64 games on a computer. Do I need another controller for the GameCube games, or can I use the N64 controller? If I do need another one, which one? And do I need another adapter?
Without knowing how exactly your Wii modified it's not possible to definitively answer your question. The Wii was originally released with GameCube support, but Nintendo removed this backwards compatibility feature late in the life of Wii. If your Wii is one of these older models then you would need a GameCube controller and probably a memory card to play GameCube games. They won't work with any sort of Wii controller. It should go without saying, but just to be clear you'll also need a GameCube game disc as well.
(While GameCube games and memory cards might be hard to find, getting a GameCube controller is actually quite easy at the moment. Nintendo recently starting selling them again so they can be used with the recently released Super Smash Bothers for the Wii U.)
It seems unlikely that your Wii was actually modified to play Nintendo 64 games. This would require cramming in a complete Nintendo 64 console inside the Wii case. More likely is that your console was modified so that it can run homebrew software and so can run a Nintendo 64 (or other) emulator. In that case the emulator should work with normal Wii controllers.
Nintendo also released a few Nintendo 64 games for the Wii as Virtual Console games. These game are purchased through Nintendo's online store and are run using Nintendo own emulator. No modification of the hardware is necessary for these game and they should work with Wii controllers. The store page for the game should spell out exactly which controllers are supported.
The answer by Ross Ridge completely covers the original question. I just wanted to add that you should look around for a Gamecube Wavebird controller. It's fully wireless and supported on the Wii. They have been discontinued for a long time but you can still find them. The button press response and the d-pad/thumbstick feel are better than any non-nintendo controller variant. You can definitely game a lot more comfortably with this controller.