Eowyn, daughter of Eomund, was Lady of Rohan. Her brother was Eomer and she was Theoden's niece.
We meet Eowyn in the second part of "The Lord of the Rings". This is how Tolkien describes Eowyn: "Grave and thoughtful was her glance, as she looked on the king with cool pity in her eyes. Very fair was her face, and her long hair was like a river of gold. Slender and tall she was in her white robe girt with silver; but strong she seemed and stern as steel, a daughter of kings." Why was there pity in her eyes? The answer is simple - she loved her uncle Theoden, who became bewitched by the spells of the malicious counsellor Grima. And worst of all - she could do nothing to help him. If it wasn't for Gandalf who came in time, who knows what would have happened to her.
When Eowyn met Aragorn she instantly fell in love with him. She managed to see in him "the heir of kings, wise with many winters". She gave all her heart to the King-to-be of Gondor, but her love was unrequited, for Aragorn's heart belonged to Arwen Undomiel.
When the power of Gondor restored, Theoden organized a campaign to Helm's deep, where the orcs besieged the stronghold of Dunharrow. He had to leave someone in charge of Gondor. And his choice fell on Eowyn, since the King's son died, and Eomer, his nephew, was the best warrior ever. "She is fearless and high-hearted. All love her. Let her be as lord to the Eorlingas, while we are gone," said Hama, the chief guard of Theoden.
But Eowyn could not wait for some outcome and do nothing at all. For she was not only a brave and fearless maiden, but a true warrior. She said once: "I am not a serving-woman. I can ride and wield blade, and I do not fear either pain or death. But I do fear to stay behind bars, until use and old age accept them, and all chance of doing great deeds is gone beyond recall or desire." She tried to dissuade Aragorn from going on the Paths of the Dead, she begged him not to do that. But all in vain.
So, in the disguise of a man named Dernhelm she left Rohan and rode to Gondor with the King's forces, sneaking Merry along at her side. Probably, it was the fate itself that willed for Merry and Eowyn to be together at the battle of the Pelennor. According to an ancient legend, one of the Nine wouldn't die from the hand of a man. So it happened - the Nazgul Lord died of the sword of Eowyn.
She was deadly injured and was also hurt by the Black Breath. During her stay in the Houses of Healing, she met Faramir, the Steward of Gondor. Soon Faramir revealed his feelings to her, however some time passed before she returned his love.
Soon Eowyn and Faramir were married, and she became the White Lady of Ithilien.