Zelda
Zelda
Like in most games, Zelda is the princess of all of Hyrule. When Ganon attacked Hyrule and took the Triforce of Power and led his army into Hyrule, Zelda took the Triforce of Wisdom and broke it into 8 pieces and told her servant, Impa, to find someone with enough courage to find the pieces and use it to destroy Ganon. But the evil king found out about this and imprisoned Zelda. Finally, Link found the Triforce of Wisdom, defeated Ganon, and saved the princess.
Zelda II
Even though The Adventure of Link happens right after The Legend of Zelda and even has the same Link, the game has a very different Zelda. This Zelda sleeps in a chamber and cannot wake up until someone finds the Triforce of Courage and uses it on her.
A long time ago, the King of Hyrule died, and passed the Triforce down to his son, the next king. But the Triforce was incomplete, missing only a third of it's power, which is Courage. A wizard told him that the only one who knew where it is was the princess Zelda. They questioned her, but she wouldn't say, so the wizard put her in a endless sleep. When the wizard died afterwards, the prince ordered every princess born in the royal family to be called Zelda, so the tragedy would never be forgotten, and thus this was her name on every single game afterwards.
A Link to the Past
When Agahnim captured Zelda, she telepathically talks to Link and tells him to rescue her from Agahnim. When rescued, he takes her to a sanctuary where a Loyal Sage lives, and leaves her with the sage to find the Master Sword. But Agahnim kills the sage and takes the princess back to the castle. She dissappears before Link and Agahnim go to the Dark World. After Link rescues the 7 maidens, the last one, Zelda, tells Link where Ganon is.
Link's Awakening
Zelda doesn't appear in this game, and thus is not an actual character, but she is mentioned once. Right after A Link to the Past, Link travels on a ship to new lands, but is shipwrecked in an island. The first person he meets there is Marin, who he confuses for Zelda.
Ocarina of Time
The Prophesy
One day, princess Zelda wakes up from a dream. In her dream, she sees dark clouds in the sky, covering Hyrule in darkness, but then a light shot out of the forest, cleared the sky, and a figure appeared. It turned into a person, holding a stone, and followed by a fairy. Since she had studied prophesies, she thought someone would come from a forest with a stone and a fairy and take down some kind of evil form.
When Zelda tells her father, he doesn't believe her, but her servant, Impa, the last Sheikah, does, and so tries to protect her. Then, a man from the dessert named Ganondorf comes and visits the king. Zelda believes that this man is the evil darkness that surrounded Hyrule in her dream. She tried to warn her father, but doesn't believe a thing, so he sends her out of the room so he can talk privately with Ganondorf. She goes into the garden and looks inside one of the windows to the Throne Room, where her father and Ganondorf are, and spies on them.
Link
Link goes through the castle guards and into the garden where he meets Zelda. She notices that Link is wearing green chlothes, like the light, and has a fairy. When she finds out that he's from the Kokiri Forest and has a green stone idenical to the one in her dream, she realizes that he is the boy from her dreams. She tells him of her dream, Ganondorf, and the legend of the Triforce. The Triforce can grant a wish to anyone who obtains it, but to get it you need 4 things: the 3 Spiritual Stones, one which Link has; and the Ocarina of Time, which Zelda has. Using these in the Temple of Time opens the Door of Time which leads to the Sacred Realm, where the Triforce is hidden. She believes that Ganondorf wants the Triforce for himself. She tells Link to find the other Stones so they can beat Ganondorf. He is led out by Impa, and Zelda waits for him to return.
The Mistake
By the time Link found the 3rd stone, Zelda couldn't wait much longer. She went to the Temple of Time, and (somehow) made some kind of vision for Link so that when he grabs the Ocarina he learns the Song of Time. With Impa, she leaves on a horse out of Castle Town, followed by Ganondorf. On her way out, she passes Link. She throws him the Ocarina of Time, but she misses and it lands in the river. She hopes that he will open the Door of Time and stop Ganondorf.
The following morning, Zelda notices the Triforce of Wisdom on her hand. She remembered part of the legend that she forgot to tell Link. When a person makes a wish out of the Triforce, it splits in 3. The part that mostly represents that person will go to him/her, and the other 2 will go to 2 other people who are worthy of using it. She believes that Ganondorf got the Triforce after Link opened the door, made his wish, and got the Triforce of Power, while Zelda got the Triforce of Wisdom.
For the next 7 years, Zelda trains to become a Sheikah, and goes as far as to change her name to Sheik. She passes herself off as a male so that Ganondorf doesn't recognize her. She/he waits until Link returns, who is trapped between time until he's old enough to use the Master Sword (which led to the Sacred Realm), and become the Hero of Time.
The Future
Seven years later, Link appears on the Pedestal of Time, all grown up and equipped with the Master Sword. Shiek appears before him, and tells him who he is, where the other sages are located, and where to get a useful item for the first dungeon. Sheik appears later in the game to teach Link the last 6 Ocarina Songs right before each dungeon: Minuet of Forest, Bolero of Fire, Serenade of Water, Nocturne of Shadow, Requiem of Spirit, and Prelude of Light (the last one is not taught before a dungeon). Sheik appears several other times throughout the game: at the Temple of Time, after the Forest Temple, where he tells Link how to go back to the past, how to rebattle the bosses (3d version), and teaches him the Prelude of Light; at Lake Hylia, right after the Water Temple, where he witnesses the lake's restoration and speaks with Link for a few seconds; and at Kakariko Village, a bit after the Water Temple, where he tries to fight the Evil Spirit of Bongo-Bongo sealed inside the Well, and teaches Link the Nocturne of Shadows.
After all 6 sages are safe, Link goes back to the Temple of Time and finds Sheik waiting for him. He tells him about the legend where the Triforce separates during the wish, and that Ganondorf has Power, Sheik has Wisdom, and Link has Courage. He also finally reveals himself to be Princess Zelda, and that she was hiding from Ganondorf by changing her appearence while she waited for Link. But, Ganondorf took this chance of capturing Zelda and luring Link into his castle.
Link goes to Ganon's Castle to get Zelda back. When Link enters the room, all 3 pieces of the Triforce resonate, which Ganondorf explain's it's due to the fact that the pieces are in the same room, and so he starts a fierce battle with Link. When beaten, Ganondorf uses the last of his power to destroy the castle with Link inside. Zelda and Link only get a few seconds to talk before the castle starts to implode. After leaving the castle, they watch as it crumbles down on itself. Link checks if Ganondorf survived the destruction, and sees Ganondorf (now transformed into pig mode Ganon) come out of the rubble. He makes a fire fence blocking Link from Zelda-and the exit-, and knocks Link's sword outside of the arena next to Zelda. After Link stuns Ganon with his other items, while the fence is down, gets back the Master Sword and beats Ganon. Zelda holds Ganon in place with magic while the sages seal him away.
Zelda has a quick conversation with Link, telling him how she's sorry she dragged him into this mess, which is all her fault. She makes the choice of sending Link back to the past so he can get his childhood back. So she asks for the Ocarina of Time and plays (in my oppinion) a better version of Zelda's Lullaby, which sends Link back to the past. Link realizes that he's right at the time before he met Zelda, meaning she doesn't know him (yet), the people he helped got their problems back, and Ganondorf is still free to walk around wherever he likes. The last scene of the game is Link meeting Zelda for the first time... again.
Majora's Mask
Zelda does not take an important role in Majora's Mask. She is only seen once, in a cutscene early in the game. After retrieving the Ocarina of Time, Link has a flashback of when Zelda gave him the powerful instrument. She teaches him the Song of Time, required to complete the game, as it allows Link to return to the first day while keeping things such as masks, items, upgrades, songs etc.
Oracle of Seasons/Ages
Zelda takes a more important role that in Link's Awakening and Majora's Mask, but it's not that big. If you connect both games for the final quest, she will appear and help you fight Ganon.
Four Swords
In this game, Zelda feels that the Four Sword is weakening, and that it's seal will break and Vaati will escape. She takes Link to investigate what's wrong. But when they get there, Vaati has already escaped and captures Zelda. After Link seals Vaati back into the Four Swords, Zelda thanks all four Links (or 2 in the Anniversary Edition: single player mode). Then, all the Links unite back into one, and he and Zelda go back and put the Four Swords back again.
The Wind Waker
Character Changes
(More info soon)
Bird Ride to Outset Island
(More info soon)
Bombs
(More info soon)
Forsaken Fortress
(More info soon)
Tetra's True Form
(More info soon)
Captured
(More info soon)
Four Swords Adventures
Much like in Four Swords, Zelda senses something wrong with the Four Sword's seal, so she goes with Link and 7 maidens to find out what happened. But then, Zelda and the maidens disappeared and Dark Link appears. Link takes the only weapon nearby, the Four Sword, and he splits into 4, but Vaati escapes and sends Link flying. The game is about Link saving the maidens and Zelda to stop Vaati.
Minsh Cap
In The Minish Cap, Zelda appears as Link's childhood friend. She gives him his shield and is turned to stone by Vaati early on.
Twilight Princess
(More info soon)
Phantom Hourglass
In this game, Zelda is Tetra again, and the pirates are in search of the Ghost Ship. They travel to the World of the Ocean King in search of it. Once they find it Tetra attempts to board it, but the ship leaves with her inside. Link tries to follow her, but falls into the ocean. When he wakes up, he finds himself on an unfamiliar island. Link later finds and boards the Ghost Ship, only to find that tetra has been turned to stone. Oshus reveals that her life energy was being sucked out of her. (which is bad since she had the Triforce). When Link defeats the first stage of Bellum, the main antagonist, Zelda wakes up. Bellum tries one more time to take Zelda, but Link defeats him again. Soon after they wake up in the Pirate Ship again, but they're friends tell them they had only been gone for ten minutes.
Spirit Tracks
(More info soon)
Skyward Sword
Zelda is a resident of Skyloft, a classmate of Link and one of his closest friends. She too enjoys soaring through the clouds on top of her own bird. Despite being a normal girl in the town, Zelda seems to have some special status amongst the people of Skyloft. Early on in the game, Zelda gives out a prize to whomever wins the flight race.
After Link successfully wins the race, Zelda leaps off of the ledge from Skyloft, dropping down onto Link's bird to congratulate him. Things eventually take a turn for the worse when a huge tornado is summoned causing Zelda to fall off her bird and get sucked to the land beneath the clouds. As she falls to the ground, she appears to be swallowed by a giant, black monster, called The Imprisoned. Ghirahim, a Demon Lord whom is trying to find Zelda for himself, claims that a servant of the Goddess rescued her from the tornado and she was taken to the Skyview Temple. It is unknown whether The Imprisoned is the servant of the Goddess, or if it is what the servant rescued her from, though it's more likely it's not the servant.
The evil Ghirahim and Link set out to look for Zelda. Link is out to rescue her while Ghirahim seeks her holiness for some unknown end. Her importance is a mystery, but she is referred to as an "oracle" and "a female servant of the goddess". She apparently has been taken to the Skyview Temple, but is taken elsewhere before either Link or Ghirahim have a chance to reach her, probably to the Earth Temple and then later to the Temple of Time. It is uncertain who will find her first and what will become of her.