If you want to store them for a long time (like if you buy a ton of Vidalias when there in season) you should store them in a cool, dry, well ventilated area with plenty of air circulation room around them. Alton Brown on Food Network actually did a show on onions. He stored his in old pantyhose, one at a time, with knots between so that you can cut one off as you use it and the rest stay there. He hung his in a storage closet under stairs. For short term, just make sure that you store in a well ventilated area and not in the plastic bags from the grocery store (that's why when you buy bulk, they come in ventilated bags).
You should NEVER, NEVER, NEVER refrigerate onions (except scallion/green onions) and you should NEVER store with potatos. Storing them together or near each other causes them both to go bad quickly.
Sorry about the soapbox, I am a foodie and an amateur gourmet chef.
HTH