We first met Éowyn, the niece of King Théoden of Rohan, in his palace in Meduseld. Frustrated by unrequited love for Aragorn and longing for the glory of battle, she disguised herself as a man, and under the alias of Dernhelm, traveled with the Riders of Rohan, carrying before her also Merry who was likewise told to stay home.
During the battle of Pelennor Fields, she fought by her King Théoden; when he was injured when combatting the King of the Nazgûl, she and Merry scrambled to help him. Confronting the Witch-king, who boasted no man could best him, she removed her helmet, exposing her long blond hair and disconcerting her adversary by the revelation that she is taking advantage of the prophesy's loophole. She then slew the Witch-king after Merry stabbed him behind the knee.
Éowyn was severly injured in this fight, and because of the poisonous effect of the Nazgûl, she faced near-certain death; however, she was treated in time by Aragorn during his brief rest in Minas Tirith. Since she didn't yet recover completely, she couldn't join Aragorn's army on their way to Mordor; however while recuperating in the Healing Halls, she met Faramir, with whom she fell in love, and after the demise of Sauron, the happily wed couple settled in Ithilien.
Note: Éowyn means "horse lover" in Old English(and, therefore, Rohirric).