Tavern on the Beach is set atop the "lobby" (actually an open-air patio) of the Marriott Morningstar resort. The restaurant itself is open-air and right on the beach - you can watch the mammoth cruise ships head out to sea and hear the waves crashing as you eat.
The service was very good for the island, and the food was good but overpriced. The shrimp and andouille sausage over cheese grits is a good choice, as is the blackened mahi-mahi. For drinks, try the Bushwhacker (I can't tell you what's in it, but after a few you won't really care). They also have occasional live music, but it's slightly cheesy (imagine Bob Marley done by a lounge singer).
To get there, drive into the Marriott grounds, then take a left down the hill to the Morningstar resort. The restaurant is the first building on the left. For an after dinner drink or two, you can walk about 50 yards to Coco Joe's and enjoy a frozen drink or a local beer from a hammock on the balcony.