If it's ultraviolet light that causes skin cancer and there are UV lights in discos, are they dangerous? If not, why?

How dangerous UV light is depends on the kind of UV light. If it's got a very long wavelength quite close to visible light then it's not much worse than blue light. Once it starts getting beyond that it's getting more dangerous and closer to x-rays. The stuff in bakers and discos is called UVA and is closer to blue light and probably isn't that bad for you.