Short answers, pulled from the story.
Théoden is the King of Rohan and Lord of the Mark in J. R. R. Tolkien's The Lord of the Rings. He appears as a supporting character in The Two Towers and The Return of the King, first as a ruler weakened by the treachery of his advisor Gríma Wormtongue, and then as a decisive ally in the war against Sauron after Gandalf restores him.
Théoden dies at the Battle of the Pelennor Fields in The Return of the King. He challenges the Witch-king of Angmar, but his horse Snowmane falls upon him, leaving him mortally wounded. His niece Éowyn and the Hobbit Merry Brandybuck avenge him by destroying the Witch-king.
The scholar Elizabeth Solopova identifies close parallels between Théoden and Theodoric I, King of the Visigoths, drawing on the 6th century historian Jordanes. Both kings die when thrown from their horses in a climactic battle, both are carried from the battlefield by weeping knights while fighting continues, and both battles are described as legendary conflicts between forces of "East" and "West."
Théoden transliterates directly from the Old English word þēoden, meaning "king" or "prince", which derives from þeod, meaning "a people" or "a nation." Tolkien used the name to create the impression that the text is a piece of recovered history, mapping the Rohirrim's ancestral language to Old English in his system of invented languages.
Bernard Hill played Théoden in Peter Jackson's Lord of the Rings film trilogy, first appearing in The Two Towers in 2002. In the films, Théoden is depicted as supernaturally possessed and prematurely aged by Saruman, played by Christopher Lee, rather than gradually weakened by poisoned counsel as in the books.
Tolkien scholars including Jane Chance and Tom Shippey contrast Théoden and Denethor as two versions of a Germanic king, noting their names are almost anagrams. Where Théoden's courage is renewed by Gandalf and he rides to battle at the Pelennor Fields, Denethor succumbs to despair and dies by suicide during the same battle. Shippey maps detailed structural symmetries across both men's stories, including each hosting a Hobbit who swears allegiance and each losing a son before the final conflict.