Short answers, pulled from the story.
Éomer is a fictional character in J. R. R. Tolkien's Middle-earth, appearing in The Lord of the Rings as the leader of the Riders of Rohan. After the death of both parents, he was adopted by his uncle Théoden, king of the Rohirrim. He later becomes King of Rohan himself.
The name Éomer appears in Beowulf at line 1959, where it belongs to a king descended from Offa of Mercia. In Old English it means "Horse-famous." Tolkien had studied the poem extensively and carried the name into his own mythology.
Éomer becomes King of Rohan at the Battle of the Pelennor Fields, when the mortally wounded Théoden names him king with his dying breath. He goes on to fight at the Battle of the Morannon at the Gates of Mordor. In 3021 of the Third Age, he marries Lothíriel, the only daughter of Imrahil, prince of Dol Amroth.
Éomer is played by the New Zealand actor Karl Urban in Peter Jackson's Lord of the Rings trilogy. Tolkien scholar Michael Drout identified Urban's helmet in the role as plainly based on the Sutton Hoo helmet, which Drout calls "the single most iconic image of Anglo-Saxon culture."
Tom Shippey describes Éomer as "compulsively truculent," contrasting him with Faramir of Gondor, who is courteous and civilised. Shippey sees this difference as reflecting the character of their nations: Gondor as "a kind of Rome," and Rohan as a simpler, more vigorous society.
In Peter Jackson's films, Karl Urban's helmet as Éomer is based on the Sutton Hoo design, which scholar Michael Drout calls "the single most iconic image of Anglo-Saxon culture." Tolkien had denied that the Riders of Rohan directly represented the Anglo-Saxons, but the helmet's design makes that identification difficult to avoid on screen.