What is the name of the fortress at Helm's Deep?

The fortress at Helm's Deep is called the Hornburg. Helm's Deep is actually the name of the valley and defensive refuge in the White Mountains of Middle-earth, while the Hornburg is the castle positioned at the head of the gorge.

In The Lord of the Rings, the Hornburg becomes the main stronghold of King Théoden and the Rohirrim during Saruman's assault. Its defenses include the Deeping Wall and Helm's Dike, but the Hornburg is the fortress itself. The name refers to Helm Hammerhand, an earlier king of Rohan whose people sheltered there during a winter invasion.

Tolkien's battle is known both as the Battle of Helm's Deep and the Battle of the Hornburg. Peter Jackson's The Two Towers presents the same distinction, although the film emphasizes the fortress as part of the larger Helm's Deep setting. Minas Tirith, Edoras, and Helm's Deep are separate places: Minas Tirith is Gondor's capital, Edoras is Rohan's capital, and Helm's Deep is the valley containing the Hornburg.