Death Note is widely regarded as THE anime for someone who has never watched any to get into the culture. Others include Full Metal Alchemist: Brotherhood, Naruto(which was the one for many otaku I've met).
Anime* you mean? Not cartoons. Most popular anime are those that have been running for a long time, or have left a great legacy. These include Dragon Ball and its sequels, Pokemon, Naruto, One Piece, Bleach, etc.
And then, there are short anime that are exceptionally good, and in some cases, have left a mark in anime. These include Death Note, Code Geass, Neon Genesis Evangelion, Full Metal Alchemist: Brotherhood, Attack on Titan.
Now, I'm not saying that these are good. But, these are undeniably popular.
Anime has appeal because it's something refreshing and new. I guess some people find it to be edgy, some find it to be mature. Also, it's versatile. Since it's animation, you can do whatever you want. And, since it doesn't have any restriction in terms of violence and sexual content.