The show is based on the video game series of the same name. Yes, I know the games have mamy different Links and Zeldas, but the TV series has just one Link and one Zelda.
Basically, Link protects Princess Zelda, her father, and the rest of Hyrule from the evil wizard, Ganon. Zelda's the guardian of the Triforce of Wisdom, and Ganon has the Triforce of Power. "Whoever gets both Triforces will rule this land forever!" So, Zelda asks Link for help, and he readily agrees. He even lives in North Castle with Zelda and her father, the King and Ruler of Hyrule. Link acts like a goofball (or maybe he really is one), and his main goal is to get a kiss from Zelda (something "always happens", though, before he can get the kiss), but Link is also a skilled fighter, and he saves Zelda a lot. Zelda is also Link's sidekick, it seems, at times. Link also has to deal with unwanted advances from Spryte, a fairie that has a crush on him and is jealous of his love for Zelda.
"Captain N: The Game Master" consisted of, I believe, 34 original episodes. It premiered on NBC on September 9, 1989. The episode length for Season 1, excluding commercials and the theme songs, was about 20 minutes.
The show is about a fifteen(?)-year-old boy named Kevin Keene who lives in Northridge, California. One night, when Kevin (played by Dorian Barag in the real-life main action sequence) is playing "Punch-Out!!" when he was supposed to be doing his homework and cleaning his room, he gets pulled through his television set (a Sony) by a huge blast of energy. His dog, Duke (played by Louie from "Critters of the Cinema"), jumps in after him.
They become cartoon characters and end up in a universe called Videoland. It appears that, in this universe, all of the characters from Kevin's video games are real.
Kevin is greeted by Princess Lana, the ruler of Videoland. For the past seven years, Mother Brain (the villain from "Metroid") has been trying to conquer Videoland. The Princess was losing the war. But then, an ancient prophecy was fulfilled, and Kevin was brought to them to save Videoland.
Kevin and Duke join the N Team, which consists of Lana, Mega Man, Kid Icarus, and Simon Belmont. Season 1 ran in the 1989-1990 season.
Season 2 of Captain N had the same episode length as Season 1, but there was a Super Mario Bros. 3 episode before and after it. The one-hour (commercials included) show was collectively called "Captain N & the Adventures of Super Mario Bros. 3" and was proclaimed Number 1 in its time slot! During Season 2, the N Team got an additional member, a supercomputer named Gameboy that looks like, well, a Game Boy!
Link and Zelda appeared in four episodes of Captain N. Thankfully, the same people did their voices. The four episodes are
"Quest For the Potion of Power",
"Having a Ball",
"Once Upon a Time Machine", and
"The Trojan Dragon".
Zelda's father has apparently died by this time, and Zelda is now the ruler of Hyrule. Link's hair is longer. Also, the two of them seem a bit more mature, and Link has seemed to stop asking Zelda for 
Kevin also meets Link, his favorite video game hero, and Princess Zelda.