Manchild: His gestures and mannerisms are still child-like when older, although it's not really his fault, since his mindset was not given time or the opportunity to age due to being asleep for seven years.
Sealed Good in a Can: When he withdraws the Master Sword, he turns out to be too young to be the chosen hero. Thus his spirit is sealed away for seven years while his body matures to be old and strong enough to fight Ganondorf the way he needs to.
Would Hit a Girl: He had to as he's releasing Jiro and his carpenters after they were all kidnapped by the Gerudo Bandits.
Younger Than They Look: While Link's body aged 7 years, his mind was frozen in time, so mentally, he's this. Ironic, as the Kokiri are typically Older Than They Look. Although, the implication is that, while a child, Link is fairly mature for his age.
Captain Obvious: Mostly due to her not being programmed to recognize the player's location when giving advice. (Like when she tells you to go to Kakariko Village even though you're already there.) However, the first dungeon has a bit where she tells you that you need a key to open a door with a chain and lock on it, which will undoubtedly prompt you to respond "No, really?" or some other variant.
There's a second Genius Bonus meaning. Navi is an old Hebrew word for prophet.
Nice Girl: The few chunks of dialogue that reference Navi (mostly the by the Deku Tree) or are her talking about herself strongly hint at her being meant to be one of these. This is in sharp contrast to every other Fairy Companion in the franchise (Manga and Cartoon counting), who all feature at least a certain degree of temperament.
The Princess of Hyrule. Despite her young age, she is wise beyond her years. She is aware of Ganondorf's plans to conquer Hyrule the entire world and she has her own plans to stop him.
Barrier Maiden: She and the other Sages seal away Ganon after Link defeats him. In the timeline where Link dies are forced to seal him in the Sacred Realm instead turning the holy place into the Dark World.
Damsel in Distress: Averted until the end for the first time in the series. Most Zeldas after followed suit, often taking a more active role in the adventure.
Hammerspace: Where Sheik's harp appears to go when not playing it.
I Never Told You My Name: Sheik lets a single "Link" slip at the end of the first "conversation' with Link in the Temple of Time despite being literally the first person to encounter him in the future and no else seeming to recognize him from when he was a child. An early indication that Sheik is someone you know.
The last of the original Sages remaining, and the Sage of Light. Rauru explains to Link that he sealed Link away for seven years in the Sacred Realm so he could age into the intended Hero of Time, that Ganondorf has taken over Hyrule, and assigns him to awaken the reincarnations of the remaining sages.
Link's childhood best friend. She is later revealed to be the Sage of Forest.
Fan of Underdog: A slight example; when Link was a kid and didn't have a fairy, he didn't have any friends in the village...except Saria herself, who (partially) raised him as a baby and when he was her age she went out of her way to be his friend when the rest of the Kokiri shunned him.
Older Than They Look: Kokiri don't age, so it's possible that Saria was older than ten years old when Link was a child. She's at least Seventeen by the Time Skip, despite still looking like a ten year old.
The chief of the Goron people. He at first rejects Link, but later comes to accept him after he clears Dodongo's Cavern of monsters. He gives Link the Goron Ruby and inducts him in as a member of the tribe. Seven years later, he attempts to rescue his people from the dragon Volvagia by going to the Fire Temple. He is revealed to be the Sage of Fire.
The Big Guy: He's not only physically imposing, but is very strong.
Last of Her Kind: The last surviving member of the Sheikah. For real, given Sheik is just a disguise.
Legacy Character: Impa is a name given to all of Zelda's caretakers in the series. Although, it was implied that this Impa is the same one who appeared in the backstory of the original Legend of Zelda.
Ninja Maid: Without the costume, but seeing that she is Zelda's nursemaid, she still qualifies.
The second-in-command of the all-female Gerudo thieves who secretly despises Ganondorf. She attempts to break into the Spirit Temple to retrieve the Silver Gauntlets, but is captured and brainwashed by Twinrova instead. She is revealed to be the Sage of Spirit.
She also refuses to kill, and is disgusted that Ganondorf is willing to.
Commander Contrarian / The Starscream: To Ganondorf, before she becomes a Sage. Zigzagged by the fact that her misgivings about Ganondorf are legitimate concerns and that her motivations aren't fundamentally evil, unlike his.
Screw the Rules, I'm Doing What's Right!: As she stated to Link, she knows that Gerudo law requires that she bow down to Ganondorf due to his being the first male born in a century to the group, but she doesn't care and will not acknowledge Ganon as her king due to the atrocities he committed.
Great Deku Tree
The guardian spirit of the Kokiri tribe. He is cursed by Ganondorf at the beginning of the game, and dies shortly after Link breaks the curse. A new Deku Tree grows in his place when Link is older, which retains all memories from the first, who tells Link that he is a Hylian and not a Kokiri.
Fisher King: Once the Deku Tree dies, Kokiri Forest goes to seed.
The wise old owl who shows up around Hyrule. He appears to give the player hints as to where to go next and what to do. However, his long-winded text conversations and Exposition Break manner of speaking made him annoying in any playthrough but the first one.
The Glomp: The Gorons love giving hugs to their friends, which freaks Link out for obvious reasons.
Proud Warrior Race Guy: The Gorons as a whole actually don't seem to exhibit this trait. While later games would see them becoming more warlike, the gorons in this game are a fairly laid-back and peaceable society who frequently need the help of a more-warlike visitor to bail them out.
Verbal Tic: They commonly tack "-goro" onto the ends of their sentences or words.
Non-Mammal Mammaries: Judging by Ruto and the Zora females seen in other games, this is typical of the race.
Sealed Good in a Can: The entire race gets frozen into the ice of their aquatic home by Ganondorf, though they probably aren't conscious for it. The only ones to escape this fate are King Zora, who's frozen in red ice and can be rescued with blue fire, and Princess Ruto, who is ultimately saved from the ice by Sheik.
A giant fish/whale hybrid. To acquire the Zora Sapphire, which he swallowed, Link must go into his belly and find both it and Princess Ruto.
The Assimilator: In Master Quest there are cows being absorbed alive into his stomach lining.
Womb Level: Uniquely in the Zelda series, he serves as not only a character, but an entire dungeon as well.
Amazon Brigade: The entire race is female, and it is required by law that they obey their king, the sole male born in 100 years.
Brainwashed and Crazy: Shortly after Link kills Twinrova, one of the carpenters would tell Link when spoken to that with Koume and Kotake dead, the Gerudo are no longer scary anymore, implying that their earlier actions were the result of Twinrova brainwashing the race or keeping them in line with fear.
Even Evil Has Standards: It's subtly implied that the Gerudo may not have actually wanted to serve Ganondorf in the adult years, as they were seen celebrating Ganondorf's defeat at Lon Lon Ranch. Granted, they weren't quite as brave about it as Nabooru, who was willing to defy him and their ancient traditions, but still.
"As I thought, you held the keys to the Door of Time! You have led me to the gates of the Sacred Realm... Yes, I owe it all to you, kid!"
Ganondorf, King of the Gerudos, and, not much later, Great King of Evil. He seeks the power of the Triforce, but, due to his unbalanced heart, causes it to split into its three basic components, and is only able to seize one of them: The Triforce of Power, the one virtue most prevalent in his heart. Of course, one third of the power of the goddesses isn't a bad share on its own, granting him tremendous physical and magical power and making him nearly immortal, and with it he easily takes over Hyrule in the seven years of Link's absence. However, just one piece of the Triforce still isn't enough to grant him his wish. And so, his search for the remaining pieces continues...
Zelda: Ganondorf... pitiful man... Without a strong, righteous mind, he could not control the power of the gods... and...
Ambiguously Brown: Him and all the Gerudo people. They live in deserts, so it's probably justified. Ganondorf himself, however, has a sickly green tone to his skin. By the end of the game, it has already turned into full-blown green skin.
Archenemy: This is where his centuries-long antagonism to many Links and Zeldas to come began.
Genius Bruiser: He's an incredibly powerful sorcerer, and also a powerful warrior. In addition, he is also a pretty good strategist. It's heavily implied that he used both Zelda and Link to access the Sacred Realm. And then years later he uses Link to reveal Zelda's location.
Magic Missile Storm: His Desperation Attack as Ganondorf is to gather a large orb of power and fire five of his energy orbs at once. Ironically this is easier to deal with than his standard attack, as Link's Spin Attack will deflect all of the bolts and at least one is promised to hit him with no chance for him to reflect it.
Nigh-Invulnerability: Granted by the Triforce of Power. You can count all weapons that were ever able to hurt him on one hand.
Although how much he resembles a pig is open to interpretation. The artwork for the game (see above) gives him a slightly piggish nose, but even his in-game model doesn't look so much like a humanoid pig as some kind of demonic, non-descript Beast Man.
Prongs of Poseidon: The Trident of Power. Though he himself doesn't use it in this game, his minion, Phantom Ganon, does.
Blade Run: Dark Link is capable of doing this if you use the stab move to attack him by freezing Link in place somehow and jumping onto the blade. This can be used to your advantage, however; if you use a broken "Giant's Knife" he falls off.
However, if you haven't figured out not to stab him before his Turns Red phase and aren't using a broken Giant's Knife (which is quitepossible), he starts punishing your idiocy and will attack you while standing on your sword.
Resurrected for a Job: Volvagia is an ancient evil that devoured Gorons alive until defeated by the Hero of the Gorons. Ganondorf revives the evil dragon to exemplify the fate of the races who opposed his despotic rule.
Curb-Stomp Battle/No-Holds-Barred Beatdown: Its first appearance has it flailing Sheik around shortly after escaping the well, and throwing the hapless Sheikah down hard enough to induce unconsciousness. The exact result of Link's failed encounter with it is unknown (the screen blacks out right when it approaches Link), although the vocal grunts imply that Bongo Bongo managed to defeat Link easily in a similar manner, and it was implied that Link was knocked out for several hours as a result.
Incidentally, you can Curb-Stomp Battle Bongo Bongo right back with the use of a glitch that allows you to drop a bomb on its hitbox before you even enter the boss arena. The dropped bomb puts it into stun mode, and you can jump down and kill it near instantly. (seen at about 4:22:00 here: http://zeldaspeedruns.com/speedruns/4541 )
Mind Control: They use this on Nabooru (with the Hypno Trinket variety). It's also implied from one of the carpenters after her defeat that the Gerudo as a whole may have been brainwashed by her as well.