This depends on what state the onions are in. Here is a guide:

Whole onions (uncut and unpeeled)
Store them in a cool, dry, dark place with plenty of air movement. The best temperature is 45-55 degrees F. They can keep up to 4 weeks when stored properly.

Peeled onions
You can store whole peeled onions in the refrigerator.

Cut onions
Store cut onions in an airtight container, bag, or wrapped in plastic wrap in the refrigerator. They should be good for up to 7 days when stored properly.