Dimitris Ammoudi Taverna is the place to visit while on the island of Santorini. Everybody talks about it and recommends it. I second that.
The absolute definition of the fish taverna by the sea right on the wave. Choose a scorpion fish and ask Dimitris to grill it for you on charcoal in foil with garlic, salt and pepper. Should you sit right below the sign of the taverna, you will place the bread basket on the marble mooring-post right by you. It has been standing there since the early 19th century, when Ammoudi was the commercial port of Santorini. Here the ships used to carry pumice. The sun vanishes behind Thirasia and from the specific spot of the bay you have the longest sunset duration!
Located in the Old port of the island of Santorini, the taverna is situated just on the water, next to the fishermen boats, at the end of a long strip of restaurants, all showing their daily fishing loot on display. Expect to eat fish, seafood and home made Greek specialties. The only bad thing is the service. The waiters are so busy that they cannot focus enough on every customer's demands.