If you love J.R.R. Tolkien's The Lord of the Rings, then you know that all of the events of the story would be impossible without the half-Elf Elrond. Elrond, the Lord of Rivendell, lived in Middle-earth from the First Age to the beginning of the Fourth Age.
The Half-elves, or Peredhil, were people of Middle-earth whose name primarily applied to Elrond and Elros in the Second and Third Age, but also to Eärendil and his wife Elwing before them. They were called so because of their mixed Edain (Mannish) and Eldarin (Elven) blood.
His parents were Eärendil and Elwing. He was thus Half-elven: Eärendil was the child of the mortal Tuor and the elf Idril, while Elwing was the grandchild of Beren (a Man) and Lúthien (daughter of the Elf-king Thingol and the Maia Melian).
Arwen is one of the half-elven who lived during the Third Age; her father was Elrond half-elven, lord of the Elvish sanctuary of Rivendell, while her mother was the Elf Celebrian, daughter of the Elf-queen Galadriel, ruler of Lothlórien. She marries the Man Aragorn, who becomes King of Arnor and Gondor.
Although he lived among them, Legolas was not fully of the Silvan Elves. As a son of the Elven-king Thranduil, who had originally come from Doriath, Legolas was at least half Sindar; his mother's identity is completely unknown.
Arwen. Galadriel's granddaughter and the daughter of Celebrían and Elrond. After the War of the Ring, Arwen marries Aragorn II Elessar and becomes Queen of the Reunited Kingdom of Arnor and Gondor. Like other Half-Elves, Arwen has the birthright to choose either immortality among the Elves or mortality among Men.
First off, Galadriel was the first cousin of Turgon, who was Elrond's great-grandfather, which made them first cousins thrice removed. Furthermore, Elrond was the great-great-grandson of Galadriel's uncle, Fingolfin.
Galadriel is one of the Calaquendi, meaning "Elves of the Light" in Quenya. Some of the earliest Elves to come into existence, the Calaquendi dwelt in Valinor with the gods during the halcyon Years of the Trees, which predate even the First Age of Middle-earth.
The Third Age. At the beginning of the Third Age, Elrond married Celebrían, daughter of Galadriel. They had three children: twin sons, Elladan and Elrohir, and a daughter, Arwen.
Celeborn and Galadriel had a daughter, Celebrían, who married Elrond Half-elven of Rivendell. During the Second Age, when the Rings of Power were forged, Galadriel distrusted Annatar, the loremaster who taught the craft of the Rings to Celebrimbor.
Why were Galadriel and Cirdan not invited to the Council of Elrond? Galadriel was tied to Lothlorien not only as the ruler of that realm, but her ring Nenya preserved the purity of that land and helped protect it from the threat of Orcs.
Originally the three Keepers were Gil-galad, Galadriel, and Círdan. Gil-galad gave his Ring to Elrond and Círdan gave his to Gandalf. At the end of the Third Age, when the Rings finally lost their power and were taken into the West, the Keepers were Elrond, Galadriel and Gandalf.
Though neither is someone you'd wish to annoy, Galadriel is generally considered more powerful than Elrond in Lord of the Rings. Not only is she older than her Rivendell counterpart, but Galadriel witnessed the light from Valinor's Two Trees, giving her a mystic quality.
However, their relationship remains purely platonic in Tolkien's source material. It's true Gandalf is in awe of Galadriel's beauty, but that isn't anything special. Everyone is in awe of Galadriel.
Aragorn was a descendant of Elros Tar-Minyatur through the line of the Lords of Andúnië to Elendil, High King of Arnor and Gondor. Like all of the kings before him, he was related to Elrond through the House of Elros.
In The Hobbit they are called the dark elves (dark because they never saw the light of Aman), the light elves (having seen the light of Aman),and the sea elves. Legolas was a dark elf, Galadriel was a light elf; so there is no relationship between them.
Like Elrond, some of Galadriel's powers stemmed from the Ring of Power that she held in her possession. Nenya, the ring of water, had the powers of preservation and concealment from evil, which Galadriel used to create Lothlórien and keep it safe from outside influences.
Galadriel was one of the Ñoldor who came to Middle-earth wishing to see the lands and rule a realm of her own. She came with the second host of Elves from Aman with Fingolfin and Finrod. In Doriath, she and Celeborn fell in love and married.
Arwen probably got to know Legolas as her husband's friend and as the leader of the Elven colony in Ithilien. As Mary Tower points out, it is possible that they are distantly related, but it's nowhere stated.
Tolkien doesn't reveal whether Legolas's mother died sometime during the Second or Third Age, or if she traveled to the West, leaving Thranduil and Legolas behind. Legolas does not marry or have any children, based on the appendix of The Lord of the Rings.
