Wii Classic
The original Wii supports GameCube games on disc, and has GameCube controller ports hidden under a plastic door.
Wii Family Edition
There is an "updated" Wii that goes by "Wii Family Edition" on the retail box. It is identical in many ways to the old style Wii, down to the form factor. However, it does not support GameCube discs or controllers.
The best way to tell them apart is the model number - the Family Edition is model number RVL-101, and the "classic" Wii is model number RVL-001. This number is printed on a label on the bottom of the unit when it is in the vertical orientation. If it's lying horizontally, this label is on the right side of the unit when looking at the front.
You can confirm this information on Nintendo's support website. Click to view the Wii manuals, and you'll see two sets, one for the RVL-001, and one for the RVL-101. The 001 manual mentions the GameCube ports, while the 101 does not.
In addition to the model number change, there are a couple of other telltale signs:
- The plastic panel covering the GameCube ports on the Family Edition is screwed down, where on the older model it flipped open. Underneath, the holes are present for the controller ports, but the ports themselves are empty.
- The serial numbers on the "classic" Wii tend to start with L, and the Family Edition serial numbers appear to start with K. However, since serial numbers are at the discretion of the manufacturer, I'd hesitate to act on that information alone.
- The Family Edition Wii doesn't ship with the vertical stand, although since the case design is identical to the previous design it can be used horizontally or vertically.
- The Family Editon's Wii logo is parallel to the disc drive slot, (making it correctly oriented when the unit is horizontal) whereas it was perpendicular to the drive slot on the original Wii.
Wii Mini/Wii U
Since introducing the Family edition, Nintendo has also released the Wii Mini in some markets, which does not support GameCube games at all. It is also missing network support, and has a top-loading disc drive rather than the front-loading drive of the original Wii design. It is essentially a bare-bones, stripped down version of the original Wii.
The new iteration of Nintendo gaming hardware is called the Wii U. Despite similar names, and cross-compatibility with certain Wii peripherals, the new Wii U does not support GameCube discs either.
Best Answer
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.