Coupons
Coupons are dollar amounts the will be subtracted from the check. Typical coupons are "$5 off a $40 purchase", or "$8 off our steak special when ordered before 5PM on Thursdays in even-numbered months". The check will usually list the item at its full (normal) price, and then have a subtraction for the coupon. The restaurant would like to know about coupons you're planning to use before they present the check.
Discounts
Restaurants off dicounts on items to entice you to visit. Discounts can be offered on different items, like just food, or drinks before 5, or on the whole check. D&D can handle discounts on either items or the check, but not both. The check will most often list the item at its discounted price. The restaurant would like to know about discounts you're planning to use before they present the check.
Gift Certificates
Gift Certificates are, to the restaurant, just like cash. When you present a gift certificate, they'll subtract its value from the amount you owe. (Since it's included in the total, you effectively pay tax on it when you redeem it.)
Totals
You may see different totals on your check. You might see a total amount before coupons and discounts are subracted and one after. The total you're interested in is the "net" total, the one that the restaurant expects you to pay. That will be after coupons and discounts have been subtracted.
If you're using a gift certificate, make sure that amount is in the total that ou enter on tab 1.