Link is the hero of the story, and the character you as the player must control. Just before leaving for the Picori Festival with Princess Zelda, Link's grandfather asks him to take a sword to Hyrule Castle that will be awarded to the champion of the Sword-Fighting Tournament. The winner of the tournament is Vaati, who releases evil upon Hyrule and turns Zelda to stone. It's up to Link to reverse what Vaati has done, and to make sure that he gets stopped from doing anything else.
Ezlo was once a Minish, but had been transformed into a "cap" by Vaati. Link meets Ezlo in his quest, and find his ability to bring them both down to the Minish-sized world particularly useful. Ezlo is a chatty and wise fella, but can sometimes be rude and self-centred.
Zelda is the princess of Hyrule, and Link's childhood friend. On the day she visits the Picori Festival with Link, she meets an unfortunate situation at the hands of the mysterious Vaati: she is turned to stone. Link is entrusted with the quest of saving her.
Vaati appeared out of nowhere, and won the Sword-Fighting Tournament with much ease. When he is presented with his prize, he reveals some of his true intentions, and turns Zelda into stone. Seldom people know about Vaati's past, but Ezlo had some connections with him previously...