Despite me knowing that the average person here isn't all that interested, I might give it a try anyway. On a related side-note, I see people have a bit of prejudice about anime, just saying that it's people with big eyes doing ninja stuff, when it's in fact quite a diversity in anime, with genres such as horror, psychological, thrillers, romance, not much unlike other stuff like regular movies and such. Well, I suppose that they big eyes are quite hard to avoid at some times, but anyway...
For you who don't know, the most common definition of anime is "animated cartoon or movie created in Japan", but as of lately, similar stuff has been created in other parts of the world, making the line between "anime" and regular "cartoon" a bit harder to distinguish.
In the "West", there are some animes that even comes into the mind of those that may not watch anime when reflecting upon it, usually based on the fact that they are a few of the ones that actually have aired there such as Sailor Moon, Naruto, Ergo Proxy, and most notably, Dragonball.
Regardless, if I were to recommend a few animes for the uninitiated based on the genre, it would probably be like this. The good thing about those, unlike some of the other animes, are that they are relatively short, ranging from 1-24 episodes, or in some cases, 1-3 seasons. I haven't watched all of them so I can't vouch for the quality of all the examples, and in most cases, they intermingle in other genres as well, and some of the titles may be misplaced as a clear definition is hard to state. Most of these have mangas (comic) as well, which often are the original work of it, thus staying closer to the intent of the creator.
Ah, those that I pick are obviously going to be biased a bit towards my tastes, so the big image is a better choice for a more neutral recommendation.
Action - Cowboy Bebop , Black Lagoon , Deadman Wonderland
Comedy - Azumanga Daioh , Cromartie High School , Sayonara, Zetsubo-Sensei
Drama - Welcome to the NHK , Mahou Shoujo Madoka Magica , Gunslinger Girl
Gambling - Akagi , Kaiji , Legendary Gambler Tetsuya
Horror - Hellsing Ultimate , Paranoia Agent , Higurashi - When They Cry
Romance - Honey & Clover , 5 Centimeter Per Second , Mirai Nikki
Slice of Life - Beck , Minami-ke , Danshi Koukousei no Nichijou
Sport - Hajime no Ippo , Slam Dunk , Eyeshield 21
Psychological - Serial Experiments Lain , Neon Genesis Evangelion , Ergo Proxy
Thriller - Monster , Perfect Blue , Paprika
For further examples, one might want to check out this chart.
For you who don't know, the most common definition of anime is "animated cartoon or movie created in Japan", but as of lately, similar stuff has been created in other parts of the world, making the line between "anime" and regular "cartoon" a bit harder to distinguish.
In the "West", there are some animes that even comes into the mind of those that may not watch anime when reflecting upon it, usually based on the fact that they are a few of the ones that actually have aired there such as Sailor Moon, Naruto, Ergo Proxy, and most notably, Dragonball.
Regardless, if I were to recommend a few animes for the uninitiated based on the genre, it would probably be like this. The good thing about those, unlike some of the other animes, are that they are relatively short, ranging from 1-24 episodes, or in some cases, 1-3 seasons. I haven't watched all of them so I can't vouch for the quality of all the examples, and in most cases, they intermingle in other genres as well, and some of the titles may be misplaced as a clear definition is hard to state. Most of these have mangas (comic) as well, which often are the original work of it, thus staying closer to the intent of the creator.
Ah, those that I pick are obviously going to be biased a bit towards my tastes, so the big image is a better choice for a more neutral recommendation.
Action - Cowboy Bebop , Black Lagoon , Deadman Wonderland
Comedy - Azumanga Daioh , Cromartie High School , Sayonara, Zetsubo-Sensei
Drama - Welcome to the NHK , Mahou Shoujo Madoka Magica , Gunslinger Girl
Gambling - Akagi , Kaiji , Legendary Gambler Tetsuya
Horror - Hellsing Ultimate , Paranoia Agent , Higurashi - When They Cry
Romance - Honey & Clover , 5 Centimeter Per Second , Mirai Nikki
Slice of Life - Beck , Minami-ke , Danshi Koukousei no Nichijou
Sport - Hajime no Ippo , Slam Dunk , Eyeshield 21
Psychological - Serial Experiments Lain , Neon Genesis Evangelion , Ergo Proxy
Thriller - Monster , Perfect Blue , Paprika
For further examples, one might want to check out this chart.
. I love that the most. More than Spirited Away which everyone goes on about. Second is Kiki's Delivery Service. 
