The Falmari, the Telerian Elves of Aman under Olwë and the Falathrim, the Elves of the Havens of the Falas in Middle-earth were masters of shipbuilding and were great mariners, but were not great sea powers and explorers like those of Men. They used their ships mainly for traveling to and from place to place and later along the Straight Road to Aman.
It was the nations of men that created great fleets of ships for the purposes of exploration and war. The Númenóreans were the first of the Mannish sea powers and they were used to colonise Middle-earth. After the Downfall of Númenór, it was passed on to the Dúnedain living in exile in Middle-earth of which Gondor was the most powerful for a time.
Pirate nations such as the Corsairs of Umbar also knew the art and used their ships to terrorize the sea and raid and plunder the coasts.