Discount codes are applied at Checkout in the "Coupon or Promotion code" field. You will see the discount applied to the total in the Checkout before you complete the order.
Discount codes differ from gift codes due to the way in which they're redeemed. The format is also different.
If your code is in the following format, XXXXX-XXXXX-XXXXX-XXXXX, see the article titled "How Do I Apply My Gift Code?"