Well of course Zelda plays a prominent role in the games--after all, they're named after her. But her role varies from game to game, and here's a quick overview of those roles in the games.
The Legend of Zelda (SNES)
Zelly-kins doesn't have much of a role in this game, except in the back story. Ganon sought the impeccable power the Triforce held so he may rule the world, and upon hearing of his insidious plan Princess Zelda split the sacred Triforce into eight pieces and sent Impa away to find a hero worthy of retrieving the pieces and reviving the Triforce before Ganon can utilize its power. When the evil prince hears of this, he imprisons Zelda, and she waits for her hero in green armor to come and save her and her kingdom.
The Adventure of Link
Zelda's role is a little confusing in this one. After the death of her father, her older brother, heir to the throne of Hyrule, was set to inherit the entire Triforce, but only received a piece of it. A menacing magician tells him that his younger sister, Zelda, knows the location of a piece of the Triforce, and when she refuses to tell him the location, the magician casts a sleeping spell on the young princess. Grief stricken, the prince locked Zelda away in the North Palace and demands that every woman from there out born into the royal family be named Zelda. These events happend many years before the ones in The Legend of Zelda; this game is a true sequel to the previous title. What makes Zelda's role confusing is that the Zelda from the first game and the Zelda from this game are two different people. Ganon's evil army is set to spill Link's blood over the ashes of the fallen Ganon, and Link must once again fight evil and revive the slumbering Princess Zelda.
A Link to the Past
Centuries after Ganondorf was sealed away by seven sages for opening the Sacred Realm and using the power of the Triforces to bring about darkness to Hyrule, evil is once again beginning to stir. Zelda, along with the maidens who are descendents of the seven sages are locked away in the castle; it's while she's imprisoned that she sends a plea to Link, a descendent of a Knight of Hyrule who assisted the sages so long ago, telekinetically. This time around, instead of waiting until the end of the game to rescue Zelda, Link does it fairly close to the beginning.