Galadriel is a fair elf, the fairest Lady in all of the Wood, as well as all of the world. When people of all races, besides orcs and other servants of the Enemy, look upon her, they fall "in love" with her. They are so taken with her beauty and mind of mystery, that in a way they fall in love with the Lady.
Many races fear Lady Galadriel, for it is said that "all who look upon her fall under her spell, and are never seen again."
Galadriel is the grandmother of the beautiful Arwen of Rivendell. Lord Elrond is Arwen's father. In the extended edition of The Fellowship of the Ring, you may have noticed that there was a long conversation between Aragorn and Galadriel, on the subject of Arwen. They were speaking of Arwen's leaving to take the ship to the Undying Lands. As you may have also noticed, Galadriel was behaiving just as a grandmother, as well as a queen, should have behaived. She said to Aragorn, "That choice is yet before her." That is a clue that she is her grandmother, but in the books it says that she is Arwen's grandmother directly.
Galadriel is the daughter of Finarfin, wife of Celeborn, and mother of Celebrian.