Oooh, this is a hard question. Not sure I can even narrow it down to one game, let alone a single aspect...

In no particular order:

2D Super Mario Games - the simplicity to pick it up and play, a good way to relax.

Sonic 3 & Knuckles - also simplistic, beautiful graphics and a great soundtrack

Halo and CoD Series - Nostalgia of all-nighters hanging with friends. Also, great soundtracks and cinematic stories.

Life is Strange - An accurate portrayal of being a teen, great graphic style, unique game mechanics, and a well-written story.

Sly Cooper series - An underrated 3D adventure game with great game mechanics that make it easy to pick up and play at any age, a fun, engaging story, and some high-brow humor.