The story begins in the realm of Angel Land, which is ruled by two goddesses: Palutena, the Goddess of Light, and Medusa, the Goddess of Darkness. Palutena loved the mortals, and would bestow upon them light to help them grow crops and make them happy; Medusa, on the other hand, rather despised the mortals, and took to not only destroying the crops which they'd worked hard to cultivate, but also to turn them to stone.
As you might imagine, Palutena was not pleased, and this is where things start to feel a tad similar to the story in a recent, popular cartoon show. The pissed-off Palutena would proceed to turn Medusa into a single-eyed, snake-haired monster before banishing her to the underworld.
Unfortunately, she probably should have gone with something more remote-- like the moon-- as Medusa managed to gain the loyalty and allegiance of the monsters who dwell beneath. Her army formed, she led the charge against the Palace in the Sky, conquering Palutena's army by turning the Centurions to stone, scattering the Three Sacred Treasures amongst her minions, and imprisoning Palutena within her own home.
In a last, desperate bid, Palutena reached out the former head of her personal guard, Pit, who was now trapped in the Underworld. She used the last of her power to create a magical bow for him to wield against their enemies, and tasked him with retrieving the Three Sacred Treasures so that he may defeat Medusa, ending her tyranny, and freeing Palutena.
Gameplay-wise, Kid Icarus is a platformer, and less a side-scroller than an up-scroller. As Pit seeks to leave the Underworld, he must travel up, up, ever up (as well as off one side of the screen, reappearing on the other). Some later portions do involve actual side-scrolling, and near the end of the game, Pit equips the Three Sacred Treasures-- the Arrow of Light, the Mirror Shield, and the Wings of Pegasus-- and goes through a side-scrolling shooter to face Medusa, who fights from within a large self-effigy of her rather monstrous appearance.
And like Metroid, the ending you receive depends on your performance; despite Pit's subsequent appearances, the best ending sees him become a full-grown adult.
Unfortunately, despite a great setting and wonderful characters, designs, etc., the game is arguably not very good. Many find it a bit too frustrating, though it's said to actually be easier as you go along, and then you can restart with your ending levels/items, making a second run even easier.
If you didn't get to play it, it is currently available on the Wii's Virtual Console. Not only that, but a remake in the 3D Classics line is forthcoming, and has already been released in Europe:
As with other 3D Classics, there are some tweaks to this one, including a new button configuration which allows for the rapid-firing of arrows, said to ease the difficulty. A group of fans have also created a 16-bit styled "Super Kid Icarus" Flash game:
Gotta love the Grim Reaper's screams, if nothing else.