Short answers, pulled from the story.
Éowyn is a fictional noblewoman and the daughter of the house of Eorl who feels trapped by her gender while her father dies fighting orcs. She disguises herself as a man under the alias Dernhelm to travel with the Riders of Rohan to the battle outside Minas Tirith on her horse Windfola.
Tolkien used Old English to represent the language of Rohirric where the word eoh means war-horse and wyn means delight. Thus Éowyn can be taken to mean Delight in horses and her name connects directly to the cavalry culture of Rohan.
She meets him while hovering near death in the Houses of Healing and her outlook on life changes completely after this encounter. She declares I will be a healer and love all things that grow and are not barren which mirrors real-world experiences of women who lost husbands in the Great War.
Jessica Yates wrote that Éowyn meets all requirements for a classic woman warrior because she possesses skill in fighting weapons and armour. She rides a horse and turns the Ringwraith's prophecy back onto him while embodying full-blooded subjectivity necessary for true peace.
Peter Jackson's films feature Miranda Otto playing Éowyn while Uma Thurman was slated for the role at one point. The Hollywood studio sent a script doctor to New Zealand with specific instructions but their advice was ignored by the director.