Kaepora Gaebora is a wise owl who guides Link throughout various games in the series. One of the Gossip Stones in Ocarina of Time (found in the Sacred Forest Meadow) says he is the reincarnation of an ancient sage. Another Gossip Stone in the same area tells Link that Kaepora Gaebora "may look big and heavy, but its character is rather lighthearted." Despite him guiding Link throughout his childhood in Ocarina of Time, he is not encountered while an adult except near the end. He is seen flying over Lon Lon Ranch when Link learns Epona's Song and in shadows when Link learns the Requiem of Spirit, but makes no other appearance until Link completes the child half of the Spirit Temple; after that, he is seen flying away during the credits. Kaepora Gaebora remarks, "Even I thought that the tales of a boy who traveled back and forth through time were merely a legend," when Link meets him for the last time. He also tells Link, "The future of all the people in Hyrule is on your shoulders. Maybe it's not my time anymore."
He also has a Terminan counterpart (though they may be one and the same) appearing in Majora's Mask, who first appears in the Southern Swamp to teach Link the Song of Soaring, and in Goron Village to help Link cross a large abyss. There are statues of his likeness spread across Termina, which have two functions: they are used as warp points that can be teleported to with the Song of Soaring, though only after they are discovered by Link, and can be used to temporarily save the game and quit.
In Link's Awakening, a similar owl guides Link, and arrives at certain points to give hints and back-story. He believes in a prophecy that says Link will wake the Wind Fish. In the game's ending, it is revealed that the Owl is a part of the Wind Fish's spirit (rather than just another part of his dream), and, once the Wind Fish awakes, the Owl vanishes.
In Twilight Princess, Link must travel Hyrule to find Owl Statues, found at points marked on the map, in order to complete the Ancient Sky Book, which was missing some of its letters.
In Skyward Sword, set before the other games, Zelda's father is named Gaepora; he bears a strong resemblance to Kaepora Gaebora.
Source: Wikipedia, "Characters in The Legend of Zelda series," available under the CC-BY-SA License.