What is the value of a 1999 Delaware state quarter?

Assuming it's from circulation, 25 cents unless it is one of a small number of rotated die errors that occurred with that particular design. The rotated die error is characterized by both sides of the coin pointing the same direction; that is, Caesar Rodney's head on the reverse points to the same spot on the coin's rim as does the top of Washington's head on the front. Don't be confused if the back seems to be "upside down", that's the way all U.S. coins are supposed to be struck - compare with another quarter of a different date if you're unsure. The rotated die error is fairly rare so it's not likely that you would have one pulled from pocket change, however.