Fun Fact: The Japanese version of Zelda II: The Adventure of Link was released before American audiences had even seen The Legend of Zelda!
Zelda II is a direct sequel to the first game. Link is now a teenager, and learns that the real Princess Zelda (apparently the Zelda from the first game is named after this sleeping beauty) is under a sleeping spell and needs Link's help. If Link can unlock six palaces in Hyrule, he can gain access to the Great Palace and recover the third and final Triforce of Courage, which will wake Zelda up.
The sequel to The Legend of Zelda dramatically changed things up. World navigation was via a top-down view, but most of the game was played like an action platformer. Zelda II also incorporated RPG features such as experience points which could be spent to level up total life, magic points and offensive striking power, and magic spells could be learned from NPCs in towns. The experience points system would never be repeated in future Zelda games on the North American market.
Link could be attacked by wandering monsters while traveling through the Overworld of Hyrule, and players could actually see the tiny portion of Hyrule that The Legend of Zelda took place in on the Overworld map in Link's Adventure!
Zelda II was well received by critics, who also recognized that it didn't carry nearly the same power as the first game.