I want to put ribbon around my cake that is covered in fondant. Is there a specific type of ribbon to use?

I like the look of all types...satin, chiffon, grosgrain. Just depends on what colour/look you want, right?

If you want it to be opaque, use satin or grosgrain. If you want it to be translucent, use chiffon.

If you are afraid that your cake may sweat and stain the ribbon with condensation (i.e. coming back up to room temp from the fridge?) then go to your local florist and buy a few yards of the waterproof coloured ribbons they use. I've done that a few times and they tend to have at least a dozen colours on hand and are willing to sell a few yards when I explain what it's for.

Putting ribbon on a fondant cake is easy. Much easier than the challenges of putting them on buttercream cakes, that's for sure. (Typically, I try to use a strip of coloured fondant as the "ribbon" on buttercream cakes instead of real ribbon. I sell clients on it as a way to assure there's no staining AND they get to eat it instead of just peeling it off and throwing it away. )