Oh dear...

Uh...first of all fans of anime don't like it called "Japanese Cartoons."
It's like calling "chopper" a "motorcycle" it's not cool.

To answer your question PP, I think the appeal of Japanese anime is it's mature story telling style, it's artwork, and the themes of many of the shows.
To put it bluntly, they're not over the top dramas, nor are they boring.
Anime is a balance of many genres mixed into what usually turns out to be a very compelling storyline put forward via state-of-the-art animation.

If you want a decent series to start off with, then I must suggest watching Space Battleship Yamato, the original Gundam, or the original Macross FIRST!
Don't jump into say Code Geass or Ghost in the Shell, or Cowboy Bebop or you might not be able to appreciate the older series due to the lower quality of animation.
The anime from the 1970s and 1980s were not made using computers like the late 80s up to now are.
Thus, the old pre-computer enhanced shows were all hand-painted, and that is what made them so incredible.

Hope this advice helps.