You may want to ask on the cacti & succulents forum, or houseplant forum. Include if the plant is growing in the ground or in a pot. Including a pic is good if you can, especially if you're not sure of the exact name of your plant.

The amount of time you need to wait will depend on the type of cactus and the size of the cut. A guess (without knowing the specifics) would be one full day for a small cutting ( I don't check this forum real regularly, but I will try to watch for a response if you can give more info or post a picture (no promises though).