The Beginning
Most Legend of Zelda games start out the same -- our protagonist, Link, is introduced as a young commoner with nothing particularly special about him. Link is normally given basic tasks to complete that allow players to get used to the gameplay, typically a "Find this and bring it to me" or "Go see this person" sort of quest. After exploring the character's environment, normally his hometown or a place he has just arrived in, something big happens that kicks off his heroic journey.
The Incidental Incident
Somehow, Link is sent into a dangerous situation that forces him on his journey. In Majora's Mask, he must confront the mischevious Skull Kid who wants to crash the Moon onto the town of Termina. In Twilight Princess, his village is attacked by monsters who kidnap some of the children. In Windwaker, Link's younger sister and other girls are kidnapped by a large bird!
The Middle
Link discovers the cause for the incident above ties into a larger scheme headed by the game's antagonist, who normally (but not always) turns out to be the evil sorcerer, Ganondorf. Link must obtain a number of magical items (pieces of the broken Triforce, pieces of a magic Mirror, magical gemstones, etc.) to reach this antagonist and be able to defeat him. The majority of the time spent in the games is learning where one of the items is, exploring a dungeon filled with enemies and puzzles, destroying the "boss" in the dungeon, and obtaining the item.
The Climax
Link has gathered all that he needs to confront Ganondorf, and heads to the location where his final enemy lies. Link frees Princess Zelda, who often aids Link in the final battle with Arrows of Light.
The Ending
Once Link defeats Ganondorf, there is a small cutscene before the game fades to its long stretch of credits. The cutscene normally consists of Link being thanked for his heroic deeds; then he returns to the life he left behind when he began his journey. There are no grand rewards, he never marries Princess Zelda -- he simply returns home after a job well done.