Ok, let's recommend my share:
Yu Yu Hakusho (112 episodes): Yusuke Urameshi is a school deliquent who always gets in trouble at school. One day he saves a kid from being run over and dies. Since he was such a badboy, neither heaven nor hell was awating for his death that day. Because of this he receives a chance of reincarnate and a "job" as a supernatural detective.
Personally, it's my favorite anime of all time. There are 2 "movies" and 4 ovas, but thei do not follow the storyline.
Jibaku-kun: Twelve World Story (26 episodes) - AKA "Bucky: The Incredible Kid": In each of the twelve world there is one "Great Child" that protects his people from the hostile beasts. Bucky is a normal kid from the First World, Primas, that have only one dream: world domination. One day, he meets Spark, the GC of primas (and the strongest of'em all), and as a matter of fate, Bucky is given the chance to become a GC. Bucky only cares about dominating the world, so Spark says he must tbe the best GC to accomplish that.
I know, it's hard to believe that it's a good story and such, but it's on #2 on my favorite list.
Magic Knight Rayearth (49 episodes): The studends from different schools on Tokyo meet by chance on Tokyo Tower and they receive a call from Princess Emerode, from Reyarth, to save her planet. Suddenly they are falling from the sky of Rayearth, are saved by a giant flying fish (?), meet a sorcerer named Clef that says their are the legendary Magic Knights, and that they must save Rayearth.
It's an anime really mixed, because it starts with a typical magic-girl formula, but ends up with a lot of sword-fighting and mechas! Theis also an OVA with 3 eps, but the events are all messed up (but it's a great OVA!).
Aria The animation / The Natural / The Ova / The Origination (54 episodes): Akari travels from Manhome (Earth) to the terraformed planet know as Aqua (formerly known as Mars) to become an Undine (a person who roars a gondola and acts as a tourist guide).
Yep, basically the primisse is that. There is no blood, no sex, no cussing, no lasers being shot out of the nose holes, no fighting. It's just a calm and relaxing journey though the days of Akari training to became a Prima (a full-fledged undine) with her friends. THIS is an anime for the whole family. Believe me, it's an great anime.
Since I'm at work right now, I can't write more, but I'll suggest some more animes, but without "introductions".
Shaman King (64 episodes)
Serial Experiments Lain (13 episodes)
Negima!? (26 episodes)
Angelic Layer (26 episodes)
Fullmetal Alchemist (51 episodes)
Love Hina (25 episodes + 3 OVAs + Love Hina Again, 3 episodes)
Death Note (37 episodes)
Rurouni Kenshin (95 episodes + 2 OVAS of 6 episodes total) AKA "Samurai X"
Also, Avatar: the Last Airbender (61 episodes) is not an anime, as it wasn't produced on Japan, but it as awesome cartoon! (or americanime, as some may call it).
There are many more, but I gotta go now
