Platform games, or platformers, a simple concept of moving your character through a side-view environment, became a hit with Donkey Kong in 1981.
Since then, many cult franchises have originated from platformers. Mega Man is one of the Japanese platformer classics that kickstarted its own franchise. Not only it looked great at that time, but it also offered freedom in choosing the level order.
Other games were more narrative-focused. Jordan Mechner's Prince of Persia became a hit (although not immediately) due to its cinematic animations but also due to its setting. A warrior in the middle east who fights through dungeons to save a princess is a classic story, after all. This genre also includes some iconic arcades such as Bubble Bobble and The New Zealand Story, as well as the Atari legend Pitfall!.
In the '90s, Disney released some fantastic platformers based on its movies, starting from Disney's Aladdin movie and Disney's Hercules. We should also not forget some Amiga legendary titles like Superfrog and Another World.
This section mainly lists "pure" platformers, where shooting is not required, or it's a secondary mechanic. If you like these games, you should also check platform-shooter.