Amon Hen was constructed in the early days of Gondor, perhaps even as early as the Second Age; its other names in Westron were The Hill of Sight and the Hill of the Eye. Aragorn says that it and its counterpart hill were made "...in the days of the great kings" (The Fellowship of the Ring) but this is open to interpretation.
The hill towers above the western banks of the Anduin, and it is one of the 3 peaks at the southern end of the long lake Nen Hithoel above the Falls of Rauros on the Anduin River, the other two hills being Amon Lhaw and Tol Brandir. The Seeing Seat was built there, close to the earlier northern borders of Gondor, serving as a watchtower for the borders of the old city of the Númenóreans.
The Company of the Ring had stopped and camped at Amon Hen after having travelled down the Anduin, and it was here that the Fellowship broke apart, with Frodo and Sam crossing the river on their way east to Mordor. This was also the place where Boromir died. Frodo sat on the Seat of Seeing and immediately he was able to see telescopically for hundreds of miles in all directions because of its magical properties.
It was at this point that Frodo made his choice to go to Mordor. Frodo now truly understood how evil the One Ring was, and knew that sooner or later it would corrupt all of his friends, as it had corrupted Boromir. In making that decision to walk alone into the land of the Dark Lord, Frodo showed that the Hobbits possessed, perhaps, the greatest courage of all.
Amon Hen was a Sindarin word that meant 'Hill of the Eye'.