Amroth was the king of Lorien after his father died in the Battle of Dagorlad (see this post for details.) He was a Sindarin elf, but he lived like one of the Silvan elves because he was very much in love with Nimrodel. She was Silvan, and greatly resented the elves of the west (Beleriand) for bringing wars into the previously peaceful Middle Earth. She loved him too, but because of her prejudice refused to marry him.
The two lived this way for many many years, until the balrog was awoken in Moria. This attracted many orcs to the area, and Nimrodel in a panic fled south towards Fangorn. Amroth followed her, and reached her at the border of the forest. There they had a “long debate”, at the end of which Nimrodel agreed to marry Amroth as soon as he brought her to a land of peace, which really only left Valinor as an option.
At the time there were still some elves living in the haven of Edhellond in southern Gondor. The couple travelled south (in some versions they were accompanied by a few friends) to the haven where they could get passage west to Valinor. At some point, though, they became separated. Amroth arrived at the haven to find that there was only one ship left, and they were preparing to leave soon. Nimrodel had not arrived yet, though, so he begged them to wait for her.
They waited, living on the ship, for some weeks. But there was no sign of Nimrodel. The other elves warned that there were many “unfriendly Men and evil things” in the mountains, and that she likely wouldn’t make it through. But Amroth insisted on waiting. Then one night there was a storm, and their ship was torn from the dock and out to sea. In despair Amroth jumped off the ship and swam for shore, but drowned in the waves.
In some versions of the tale it says that Nimrodel finally reached the havens, but long after the ship had left, and after Amroth had drowned trying to return to her. Of her ultimate fate nothing is said, but the region later came to be known as Dol Amroth.
SOURCES: The Unfinished Tales