'My friend,' said Gandalf, 'you had horses, and deeds of arms, and the free fields; but she, born in the body of a maid, had a spirit and courage at least the match of yours. Yet she was doomed to wait upon an old man, whom she loved as a father, and watch him falling into a mean dishonoured dotage; and her part seemed to her more ignoble than that of the staff he leaned on.
Éowyn was as brave and as fierce as her brother, and perhaps as skilled with a blade as her brother also : her father Éomund, who had to deal with Orcish invaders each day, probably learnt both his children how to wield a blade and how to ride a horse as soon as they had outgrown their diapers ; furthermore, after their father had died and Théoden took them in his household, Éowyn was probably further educated in the art of combat together with her brother, possibly by Théoden or Théodred themselves, as they probably noticed how their young niece loved to fight and was almost as skilled as her brother. Not to mention, it probably was custom that every man and every woman in Rohan would at least learn the basics of swordfighting and horseback riding, as both Mordor and Isengard were rearing their ugly heads in those days.
Yet, as Gandalf points out in the above quote, Éomer could have the time of his life in Rohan, whilst Éowyn could not, because she was a woman and therefor it was expected that she stayed in her bower in Meduseld all day, sewing and gossiping with the other ladies – needless to say, this was not the life Éowyn wanted, she wanted a life of action like her brother, she wanted to do great deeds in the spirit of her great forefathers. Hence, the life she led didn’t satisfy her, to say the least, and she was restless and unhappy because of this, and felt like she was trapped in a cage, “all her life seemed shrinking, and the walls of her bower closing in about her, a hutch to trammel some wild thing in”, like Gandalf put it.
When Théoden became completely overcome by Gríma’s poisonous words, Éowyn was forced to take care of her debilitated uncle, a task she undoubtedly found embarassing and inferior (even though she loved her uncle like a father), and completely the opposite of the ‘great deeds’ she imagined herself to do one day.
But then suddenly Gandalf, Aragorn, Gimli and Legolas came to Meduseld and awoke Théoden from his lethargy, after which the latter then continued to restore the glory of Rohan – and when Éowyn first met Aragorn, she was greatly impressed by his barely concealed greatness and glory, and felt that he was a man who had done great deeds, and who was about to do even greater deeds, and felt attracted to him as she felt she could finally know glory if she was on his side.
As she stood before Aragorn she paused suddenly and looked upon him, and her eyes were shining. And he looked down upon her fair face and smiled; but as he took the cup, his hand met hers, and he knew that she trembled at the touch. 'Hail Aragorn son of Arathorn!' she said. 'Hail Lady of Rohan!' he answered, but his face now was troubled and he did not smile.
No, at this point she did not love Aragorn, she only loved the glory he could lead her to, she loved the vision she had of him – she was attracted to Aragorn, but not more.
Only later in the end, she loved Aragorn, but not like one loves a lover, she loved him like the people of Minas Tirith loved their King.
After the events described above, and after the battle of Helm’s Deep, Éowyn was obviously very proud that the glory of Rohan was restored, and that her people were finally making war on Sauron and Saruman, even though she didn’t take part in it. At this point, Théoden had made her responsible for the people of Rohan in his absence, and she would have taken up this responsibility upto the end, she even found peace, if something dramatic hadn’t happened : the Passing of the Grey Company.
When she saw Aragorn again for the first time since he came to Meduseld, she probably thought that he was going to ask her to join him in battle (the folly of youth), but when it turned out he was taking the Paths of the Dead, which sounded like mere suicide in her ears, all confidence and peace she had known immediately went down the drain : the man she felt attracted to, who she imagined she loved, who she thought would lead her and others to glory, had chosen to throw away his life and she was completely unable to deal with this. In complete desperation she offered to join him, which he of course refused as she had a responsibility to her people. After he had left her, she thought he had died and after that she decided for herself that she’d die in battle as well, as she felt her life was again empty – the only beacon that had come to lend some of its light in the darkness which was her life, Aragorn, was gone, and perhaps she also felt that Aragorn was the only hope to save Rohan, and now that he was gone everything was lost anyway and she wanted to die a warrior’s death, instead of waiting to be struck down with the rest of the Rohirrim women, elderly and children that stayed behind.
The course of events after that is well known to everyone : Éowyn made it to the Pelennor Fields as Dernhelm, together with Meriadoc she defeated a powerful enemy, and was afterwards healed by Aragorn. When she woke up after her disease and thought about the course of events, she was both proud and embarrassed : proud as Théoden had died a glorious death and the Éorlingas had fought bravely and valiantly, yet embarassed that now everybody would know that Aragorn refused her pleas at Dunharrow, after which she tried to run into certain death like a lemming off a cliff. Yet, she was wrong in thinking this, as everybody in Minas Tirith saw her as a brave woman and everybody in Rohan regarded her as a true heroïne, and gave her the title "Lady of the Shield Arm”.
Yes, Aragorn had cured her body, but it took Faramir to heal her spirit, to finally release her from her gloom and restlessness. Or maybe when she met Faramir her restlessness ended as she finally found what she had been looking for her entire life : something that would give her life meaning and give her happiness ; she had always thought this something would be glory in battle, but in the end it turned out it was love.