Like her own role in the Zelda games, Zelda's familial background changes with each title. Here's a brief overview of some of the members of her lineage.
King of Hyrule
The much loved and kind-hearted King of the kingdom of Hyrule, and father to Zelda. He's appeared in many games, but his most prominent roles were in The Adventure of Link and The Wind Waker (under the guise of The King of Red Lions). He's gone by many names, such as Daphnes Nohansen Hyrule (The Wind Waker), and Daltus (The Minish Cap). In every title he is portrayed as a wise old man who is able to wield the power of the Triforce for the good of his kingdom; unfortunately that makes him an obvious target for such villainous Hyruleans as Ganondorf.
Prince of Hyrule
Zelda's (an ancestor of the frequently appearing Princess Zelda) older brother and who only appeared in The Adventure of Link. Apparently, after their father's death, the Prince was set to inherit the Triforce; however he only inherited a fragment of its power. When he was searching for the Triforce he came upon an ill intentioned wizard who passed on a piece of juicy information: Zelda, his beloved little sister, knows the location of the Triforce. He angrily questions her, and she refuses to give up any information. In turn the wizard cast a sleeping spell on Zelda. The Prince was instantly remorseful for the turn of events, and ordered that every female born into the royal family be named "Zelda".
Tetra's mother was once the leader of a band of pirates, just like her daughter. She was notorious on the seas, and made quite a name for herself and oversaw many treasure-seeking adventures. By the time of the events in The Wind Waker Tetra's mother has died, but her daughter has assumed the role of pirate queen with gusto.
Zelda's father in Skyward Sword, which took place many, many years before the main events of The Legend of Zelda (meaning that the Zelda featured in this game, along with the other members of her family, have yet to hold a monarchy over Hyrule). He is the headmaster of the Knights Academy in Skyloft, and wields an incredible amount of wisdom and knowledge.
An ancient King of Hyrule and likely the oldest recorded King in history. He's an ancestor of Zelda's, and seen in the game The Minish Cap where he tries to aid in governing his people from beyond the grave.