In the book and the movie The Two Towers, Éowyn and Aragorn met when he arrived in Rohan with Gandalf, Gimli, and Legolas. She fell in love with Aragorn, and confessed her feelings for him in The Return of the King.
Éowyn was initially intended to marry Aragorn. Tolkien considered that Aragorn loved Éowyn after her death, when she avenged the death of King Théoden. However, he later abandoned the concept, as he considered Aragorn to be "too old and lordly and grim". Éowyn was also once the twin-sister of Éomund.
Aragorn fell in love with Arwen at first sight. Thirty years later, the two were reunited in Lothlórien. Arwen reciprocated Aragorn's love, and on the mound of Cerin Amroth they committed themselves to marrying each other.
Éowyn falls in love with Aragorn, but though he respects her, he does not return her feelings, as he is betrothed to the elf Arwen. As Aragorn points out, her duty is with her people; she has to shoulder the responsibility of ruling Rohan in Théoden's stead when the war-host of Rohan go to war.
And in another draft, Eowyn perished after avenging Theoden during the Battle of Minas Tirith, which led to Aragorn finally admitting his love for her after she was gone.
In short, Éowyn made an impression on the Ranger from the moment he first saw her. It may not have been for long, but there was certainly an attraction from the beginning on Aragorn's part. Although he did reject her affections eventually, it may have been out of a sense of duty rather than Aragorn's personal feelings.
When Aragorn leads his company away from Edoras towards the Dwimorberg, the haunted mountain, and the Paths of the Dead, he leaves Éowyn behind him, his last words to her nothing more than, “Nay, lady”.
It is later mentioned that Tauriel is a talented warrior and was therefore made leader of the Mirkwood border guards. Legolas, the son of Mirkwood's Elven king Thranduil, is indicated to be attracted to her, but as she is a lowly Silvan Elf, she does not believe herself worthy of him.
Her broken arm was set by the healers, but they could not cope with the Black Breath illness inflicted by her contact with the Nazgul. Aragorn was able to heal her using athelas, with the mystical assist of his royal blood to seal the deal.
Galadriel and Celeborn had a daughter named Celebrian, who married Elrond and became the mother of Arwen (as well as her brothers, Elladan and Elrohir).
And when they later meet at Rivendell, the two clearly have a long history together stretching long before joining the Fellowship. This isn't surprising, as, despite their looks, Aragorn is 87 during this era, and Arwen is over 2700 years old, so they've had more than enough time to know each other.
They married in 3020. The answer can be found in HoME 12: "He wedded in 3020 Éowyn sister of King Éomer of Rohan." "It is said also that in 3020 Éowyn Éomund's daughter wedded Faramir, last Steward of Gondor and first Prince of Ithilien, in the king's house of Rohan."
After killing the Witch-king, Eowyn fell ill with the Black Breath, due to a combination of the injuries she had sustained in battle, as well as her grief over Théoden's death.
Uma Thurman turned down the role of Éowyn in The Lord of the Rings films because she had just had her first child and didn't want to move to New Zealand for several years of filming.
The Lord of the Rings films showed Arwen (Liv Tyler) dying due to her connection to the One Ring. The closer Frodo Baggins (Elijah Wood) and Samwise Gamgee (Sean Astin) get to Mount Doom, the closer she gets to death.
Legolas does not marry or have any children, based on the appendix of The Lord of the Rings. Regardless of who his mother was, Legolas is the last of his line. When he sails into the West at the beginning of the Fourth Age, he leaves no family members behind.
Legolas and Gimli's relationship is strong in The Lord of the Rings, even though it wasn't explicitly romantic in Tolkien's source material. The members of the Fellowship in The Lord of the Rings forge an unbreakable bond while on their mission to destroy the One Ring.
In the year 121 of the Fourth Age, after Aragorn's death, Arwen died of a broken heart at Cerin Amroth in Lórien, and was buried there one year after the death of Aragorn, to whom she had been wedded for 122 years. She was 2901 years old.
In The Two Towers, while talking with Eowyn, Aragorn casually mentions that he's actually 87 years old. When Aragorn is off running for miles and miles and fighting orcs and trolls and Uruk-hai and doing all his other Lord of the Rings adventures, he hardly behaves like a regular human 87-year-old.
He then tells Aragorn that the Orcs had taken Merry and Pippin before passing away, with Aragorn, Legolas and Gimli left to send his body off down the river. There is no heart-wrenching final speech, and the very last thing Boromir says is, "Farewell, Aragorn! Go to Minas Tirith and save my people!
I cannot give you what you seek." Aragorn: "I have you wished you joy since first I saw you." [He touches her face then turns away.] [Éowyn is shocked and saddened.]
The Maia Olórin took the form of Gandalf on Middle Earth around the year 1,000 of the Third Age, and Legolas was presumed to have been born some time during the Third Age's first century.
Eowyn did not quite make it to “dead”, so Aragorn SAVED her life but did not “bring her back”. Since her and Faramir (and presumably some others) were suffering from the Black Breath rather than a full-on Morgul Wound, his application of athelas was enough for them to recover from their condition.
