Hero's Spirit

In Twilight Princess, the Hero's Spirit, also known as Hero's Shade, is an undead specter that taught Link each of the seven Hidden Skills throughout the game. The Hero's Spirit is also believed by some to be the spirit of Link from Ocarina of Time.

The Golden Wolf
The Golden Wolf, also called the White Wolf, is the animal incarnation of the Hero's Spirit. The wolf is summoned by finding a Howling Stone, with the exception of the first time, in which he confronts Link directly. Once a howling stone is found, the song has to be howled in tune with the Spirit. He then tells Link to take "sword in hand" and find him, implying that Link had to return to human form before searching for the Wolf. Once Link encounters the Hero's Spirit, the Golden Wolf will appear as a symbol on Link's map. When he confronts the Wolf, Link is transported into an ancient arena. The wolf then finally transforms into his true form, the Hero's Spirit, who then teaches the young hero one of the seven hidden skills. The true form of the Hero's Spirit resembles the spirits in Hyrule Castle's graveyard.

Hero's Lineage Theory
One of the most common discussions about the Hero's Spirit is the possibility that he might be an ancestor of Link. Some of the facts proving and disproving this theory are below. Another good source of information for this theory can be found here.

Reasons both supporting and yet questioning that the Hero's Spirit is an ancestor of Link include:


 * 1) At the end of his teachings, the Hero's Spirit said that he accepted the role as the hero long ago. This implies that he may be an earlier ancestor of Link. This theory is also supported by the Hero's Spirit saying the following to Link: "Go and do not falter, my child!" Though this in itself is not conclusive evidence, since many mentor-like characters in several stories refer to their young charges using the phrase "my child," it does serve to leave open the possibility of Link being his actual blood descendant.
 * 2) Further more, evidence of the Hero's Spirit being an ancestor of Link is given just prior to learning the Mortal Draw. The Hero's Spirit said that the skills he taught Link were forgotten ways that "do not leave our bloodline" . He implicitly used the word "our" instead of "my". This clearly indicates that Link and the Hero's Spirit are indeed related in some way.
 * 3) The Hero's Spirit was once one of the many great heroes of Hyrule . It is still not known if his name was once Link as well, or perhaps even is the same man who became the Hero of Time. However, one thing that supports he may be the Hero of Time is that most of the songs used to summon the Golden Wolf are from Ocarina of Time and Majora's Mask, but why the Ballad of Gales is included is unknown (possibly because the Hero of Winds, who is revealed to be the Hero of Time reincarnated near the end of The Wind Waker, learned this song amongst others in the other branch of Hyrule's history, and though they do not occur in the same timeline, they are linked to the Hero of Time regardless.)
 * 4) In regards to physical appearance, the way the Hero's Spirit moves his feet is similar to the way Link moves his feet in Ocarina of Time when he targets an enemy.
 * 5) Lastly, in both Wii and GameCube versions of Twilight Princess, Link and the Hero's Spirit both wield their swords in the same hand. The GameCube version (the version in which the game world was originally meant to be portrayed) shows Link to be left-handed. In every past game of the Zelda series, Link's sword-hand has always been his left one, whereas there's been no other left-handed person to date in a Zelda game except in the Wii version of Twilight Princess.
 * 6) It is also worth noting the blade that the Hero's Spirit holds whenever he's seen in-game. The hilt is completely unrecognizable, although, along with the armor he wears, it appears archaic and much older than most of Link's gear was in Twilight Princess, or Ocarina of Time. However, the blade bears similarity to the shape of the Master Sword. They are both sharpened on inclines towards the edge of the blade. However, the tip of the Hero's Shade's sword is more concave on either side, and the part of the blade where the hexagonal part of the Master Sword would be is a more rounded shape, with a different design in the middle. Granted, the blade is worn and tarnished. Whether this is purely symbolic or is indicative of the Shade's true identity is still left undefined, but this resemblance may imply that perhaps the Hero's Spirit was, in life, one of the few wielders of the Master Sword.

Trivia

 * The Hero's Spirit bears some similar resemblance to Igos du Ikana from Majora's Mask. He shares the stalfos appearance with the Knights of Hyrule that were twisted by Ganon's dark magic and transformed into stalfos in Four Swords Adventures. He also teaches Link his sword techniques, like the Hyrulean Knights do in The Adventure of Link.
 * An interesting fact is that the first song is the Song of Healing, which could possibly mean that Link, similar to the way he did in Majora's Mask, heals the spirit's soul of regrets for not teaching his Hidden Skills to another hero when he was alive, and enabling him to teach them to Link.
 * Once Link learns the final Hidden Skill, the screen goes white and the Hero's Shade speaks his final sentence, "Go, and do not falter, my child!". This is similar to how the Seven Sages from Ocarina of Time speak their final words to Link.
 * The Hero's Spirit's armor bears a resemblance to the Magic Armor in Twilight Princess.