Opinions differ as to which game is the best, but A Link to the Past and Ocarina of Time are frequently declared as such.
The Adventure of Link is generally considered the worst, and has relatively little in common with the rest of the series in terms of gameplay.
The Legend of Zelda (1987) was one of the classic great games created by legendary game designer Shigeru Myamoto. The game was inspired (oddly enough) by the oft-maligned Ridley Scott film [Legend]. It began the story of Link, a boy who must gather pieces of the Triforce, which Princess Zelda had scattered across the land of Hyrule to prevent the evil Ganon from becoming all-powerful. The gameplay was controlled from a top-down perspective and involved a lot of free-roaming across a large overworld, and exploring a series of maze-like dungeons, each of which contained a piece of the Triforce, a new gadget for Link, and a fearsome boss. This formula set the stage for all future Zelda games.
The Adventure of Link (1988) continued the story, and brought some radical changes to the gameplay, most of which were immediately discarded in subsequent entries. In this one, Link had grown up, and an evil wizard placed the Princess Zelda under a sleeping curse. Treking out into Hyrule again, Link attempts to place the magical stones neccesary to break her curse and confront the evil wizard. While the world map utilized the traditional overhead view, all combat and dungeoning was done from a side-scrolling perspective. The gameplay involved 2D platforming and fighting, and also had a FinalFantasy-esque leveling and magic system. This was also the first entry to introduce Link's doppelganger, "Dark Link", as well as the only game (IIRC, someone change this if I'm wrong) where Link and Zelda share a romantic kiss (of course, in most of the others they're too young).
A Link to the Past (1991) returned to the style of the original, with an overhead perspective, child protagonists, and Ganon as the main enemy. However, rather than being about the Link from the NES games, this game told the story of a previous Link. This Link must also stop Ganon from acquiring the Triforce, although this time not by gathering the Triforce itself, but by entering Dark World to recover the descendants of the seven sages who locked the Triforce away in the Sacred Realm to begin with. The game had more in-game plot than its predecessors and also introduced a variety of Zelda standards, including the ever-popular hookshot. It also put much more emphasis on puzzle-solving as an element of the dungeons, which was later carried over into later Zelda games, as well as recent titles such as Star Fox Adventures and Golden Sun.
Link's Awakening (1993) takes place right after the story of A Link to the Past, and has Link setting out on an ill-fated voyage across the seas which lands him on a mysterious island. The game played a lot like Link to the Past, but modified the interface to accomodate the lack of buttons on the GameBoy. The game introduced Roc's Feather (allowing the player to jump), which returned in later 2D Zeldas, and contained heavy cross-overs to Super Mario Brothers. Also notable is the surprise ending, and a bit of censorship in the US version which modified a sidequest that originally had Link searching for a mermaid's bra.
Ocarina of Time (1998) is often hailed as [the greatest game of all time], and most gamers would probably agree that it's at least a worthy candidate if nothing else. While Ocarina recycled elements from all of the past games (such as Dodango from the original, Dark Link from Zelda II, Lightworld/Darkworld from Link to the Past, and the Mario cameos from Link's Awakening), the game found its own ground by creating a unique and compelling system to convert the experience into 3D, which of course has been copied by many other games since then. The plot began with yet another incarnation of Link, who has been called by the divine protector of his village (who's a tree, incidentally) to aid young Princess Zelda in thwarting the machinations of Ganondorf, the lord of thieves (name resemblence to "Ganon" is not coincidental). The game introduced many new ideas to the world of Hyrule, such as the Gorons and the musical ocarinas, and contained a variety of interesting plot developments that I won't spoil (although SuperSmashBrothersMelee reveals a big one). Also notable for having some really skanky Great Fairies.
Majora's Mask (2000) is best described as a side quest to Ocarina of Time. While it's shorter (and generally less loved) than its predecessor, Majora's Mask did some very interesting things in terms of gameplay. During the course of the game, the player gains three masks that allow him to take the form of three species present in Hyrule (and if you're really awesome and get all the masks in the game, there's a fourth mask that can only be used in boss fights because it's really broken). In addition to Link's new forms, the game takes place over the course of three days (think Groundhog's Day x 3). At the end of the three days, you have to start over. While you keep all your possessions, the world reverts to the state it was in at the beginning of the game.
Oracle of Seasons/Oracle of Ages (2001) continued with the same Link as the NintendoSixtyFour? games, this time sending Link into other worlds in which he must command time and seasons in order to stop the witches Twinrova from resurrecting Ganon. The games were odd in that while each one was a full adventure, you had to beat both to get to the final boss fight. In addition, the second game you played would, via a password system, turn into a sequel for the first one you played, and you would occasionally be sent back to the first game to retrieve some stuff. Both games played similarly to A Link to the Past, although they brought enough new ideas to the series to be considered original in their own rights.
The Wind Waker (2003) was the first (and only, to this date) Zelda game to be done with cel-shaded graphics, which caused a thousand wankers to scream "TEH KIDDIE!" on message boards across the world. Of course, true aficionados of the series know that Link has usually been a kid in his adventures (Ocarina of Time and Zelda II being the exceptions that featured teenage Link). The Wind Waker followed in the footsteps of Ocarina of Time, but featured a drastically different world design (an archipelago), and thus involved much sailing around. The story, while it starred yet another version of Link, connected heavily to the Link from Ocarina of Time and explained the tragic events that led to this new vision of Hyrule.
Four Swords Adventures (2004, and the associated Four Swords add-on to the Link to the Past GBA variant) is designed specifically for multiplayer Link action. While the Gamecube version can be played solo (and in fact is very much fun, and in EvilSouthie's opinion, worth the price of the game by itself), the game truly shines when you get four people to link four GBAs to the gamecube ports and each control an individual Link. Both Four Swords games are "cooperative" in that there's a total score that has to be reached by the end of the level, but since there's also awards for receiving the individual highest score, competition gets pretty fierce. The playstyle and graphics style are very much based on Link To The Past, using effectively the same "world". The enemy is not Ganon this time around, but instead a wind mage named Vaati. Various puzzles around the worlds will depend on being able to use more then one player at once (lifting heavy rocks, pushing heavy switches, standing on multiple triggers), and the cooperative-competitive balance is amazing. The GBA version manages to randomly create the layout of the levels with puzzles based on how many people are playing, so the game is "new" each time you play through it, and even then there's added difficulty levels depending on how many times you've played through it. The Gamecube version adds the element of the players being able to play on either the TV or the GBA connecting them to the gamecube depending on where they are and how close to the other players they are. The utter dread that anybody who's played Link To The Past will feel when they see the army of twenty blue soldiers (on the first level, no less) sprinting towards the player on a limited-room bridge is intense.
The Minish Cap (2005) is a short but sweet entry into the Zelda series for the GameBoy Advanced. Basically, Zelda gets turned to stone by Vaati, and Link must make contact with the tiny Minish people in order to forge the master sword and break the spell. The main gimic of the game is that, via Minish magic, Link will shrink and become a few inches tall, which enables him to slip through tiny holes and talk to animals, but also makes a step an insurmountable obstacle and a stray cat a deadly menace. Following up on many of the concepts introduced in Four Swords, the game also includes a lot of puzzles that require Link to temporarily produce mirror images of himself. Other notable aspects include a fantabulous sky battle, a talking hat, and a dungeon item that is pretty much an industrial strength vacuum cleaner.
Twilight Princess (2006) is probably best described as "Ocarina of Time on steroids".
Skyward Sword (2011) was announced at E3 2010. It relies heavily on motion of the Wiimote and Nunchuck, and uses the Wii Motion Plus. The live demo at E3 had interference problems, but people demoing on the floor said the controls feel very natural. It is rumored to be the prequel to Ocarina of Time, though since the Master Sword is forged sometime in the game it would make this game first chronologically in the entire series.