A sale is when everything in a shop is reduced. A discount is some money taken off the price of the item. Normally, there is always an exception.

It depends on the rest of the sentence. If you are describing where a person is located, then "he is at a sale." A person cannot be "at a discount." If you are describing that an item is being sold for a lower price, then "it's being sold at a discount" or "it's on sale." An item cannot be just "it's at sale."

A discount means that you save a certain amount or percentage off of the "normal" price of the items. A sale means you get a discount on the items that are on sale. Theoretically a sale is a short-term discount, but some unscrupulous retailers have "sales" that last forever!