This guy's the Lord of Lothlórien, whom we meet in The Fellowship of the Ring. In The Two Towers Merry asks Treebeard why Celeborn warned them away from Fangorn, and Treebeard explains that Fangorn is a strange place, and people do run into troubles. Treebeard also clarifies: Lothlórien means "Dreamflower," but it was once called "Land of the Valley of the Singing Gold," Laurelindórenan. Treebeard meditates that the change of name means that the Golden Wood is "fading, not growing" (3.4.58).
Eärendil carries one of the three Silmarils (special, shining Elvish jewels) across the sky as the Morning and Evening Star. It is the light of this star that Frodo carries in the jewel Galadriel gave him to bring light to dark places. When Frodo first holds the light above his head in Torech Ungol, he cries out "Aiya Eärendil Elenion Ancalima!"—"Hail, Eärendil the brightly shining star!" (source, pg. 278). For more on Eärendil's origin story, check out our "Character Analysis" of Eärendil in The Fellowship of the Ring, or, of course, The Silmarillion.
At Cirith Ungol, as Sam battles Shelob, he suddenly speaks a hymn to this figure as though he is speaking in tongues. This is Elbereth, the Elvish Queen of Stars. We have seen signs of the strength of her name against evil before, in The Fellowship of the Ring Book 1, Chapter 11 when Frodo tries to fight back against the Nazgûl at Weathertop. But this is the first time that we see her power almost possess someone, since Sam's "tongue was loosed and his voice cried in a language which he did not know" (4.10.9). After he has spoken this hymn, he stands and is "Samwise the hobbit, Hamfast's son again" (4.10.9). And it works: the hymn, combined with Galadriel's jewel, makes a light that sends Shelob crawling back to her lair in agony.
Fëanor only comes into The Two Towers in passing; his story really appears in The Silmarillion, if you want to know more about him. His main importance to this story is that he is the creator of the seven palantíri, one of which belonged to Orthanc. It is through the palantír, which is a glass ball that communicates with the other six palantíri, that Saruman and Sauron managed to stay in touch.
Galadriel is the beautiful elf Lady of Lothlórien, who we see a lot of in The Fellowship of the Ring. After getting back from his almost-death in Moria, Gandalf meets up with Galadriel to hear all the news of what he missed while he was dead. While we don't really see much of Galadriel in The Two Towers, she still influences Our Heroes. Galadriel uses Gandalf to pass on messages to Aragorn, Legolas, and Gimli, telling Aragorn to beware of the Dead, Legolas, of the Sea, and Gimli, of putting his "axe to the right tree" (3.5.150).
When Sam tries to describe Galadriel to Faramir, he says that she is sometimes "like a great tree in flower, sometimes like a white daffadowndilly, small and slender like. Hard as di'monds, soft as moonlight. Warm as sunlight, cold as frost in stars" (4.5.128). Sam can only describe Galadriel's beauty with a bunch of opposite ideas because her gorgeousness is beyond words. Galadriel is the ideal of perfect beauty, different for each person because we all have changing ideals.
Sam also tells Faramir that it was under the gaze of Galadriel that Boromir discovered his own desire for the Ring: "it's my opinion that in Lórien he first saw clearly what I guessed sooner: what he wanted. From the moment he first saw it he wanted the Enemy's Ring!" (4.5.132). It makes sense that Galadriel's special magic object in The Fellowship of the Ring (Book 2, Chapter 7) is a Mirror. She herself is like a mirror, which shows you your own innermost heart (either good or evil). Boromir may not have liked what she reflected back to him, but that's on Boromir himself. After all he's the one who felt guilty just looking at her.
Also, in The Two Towers we are reminded that Galadriel is the one who gave Frodo a special jewel that glows with the light of the Evening Star, Eärendil, near the end of The Fellowship of the Ring. In this novel, Frodo clutches this jewel when he freaks out at the sight of the Lord of the Nazgûl outside of Minas Morgul near the end of Book Four. He and Sam also use it to move through Shelob's Lair on the way to Cirith Ungol. So thank you, Galadriel.
Haldir is an elf of Lothlórien. In The Fellowship of the Ring, he presented Sam with some elvish rope, "a bit thin, but it's tough; and soft as milk to the hand" (4.1.64), as Sam tells us in The Two Towers. Sam recalls Haldir's words as he is using the rope to save Frodo from a cliff ledge in Emyn Muil: "[The rope] may be a help in many needs [...] And he spoke right" (4.1.60). It's a passing mention; for more on this character, you should check out our "Character Analysis" in our The Fellowship of the Ring learning guide.