I'm a big fan of manga (Japanese for comics, and often used by English speakers like me to mean Japanese comics specifically) and anime (Japanese animation), to the point where nowadays most of my comic-book purchases are manga, and most of my DVDs are anime. What is it that I like so much about it? It's a bit hard to say, but the variety is certainly part of it. Manga and anime are much, much bigger in Japan than the equivalent in America (the top selling weekly manga magazine has a circulation greater than Time magazine does in the US). In Japan everyone reads manga, and everyone watches anime; it's mainstream the way television is mainstream.
As a result, there's a much wider variety of genres and approaches to manga and anime; instead of being limited to superheroes (American "mainstream" comics*) and comedy (American cartoons), there are all kinds of things for all kinds of tastes: office life (OL- Office Lady), romance (Maison Ikkoku), James Bond-style adventure (Lupin III), Hard SF (Cowboy Bebop), crime drama (Strain), Horror (Mermaid's Scar, Parasyte), etc. And that's not even getting into the niche stuff that has never been translated into English (mah-jong comics, baseball comics, sushi-chef comics, fishing comics, and on and on).
* I know this is an over-generalization, there are plenty of great non-superhero American comics, running the gamut from A (the amazing Amy Unbounded) to A (the superlative Akiko), but I'm painting with broad strokes here. Overall I the proportion of manga to American comics I buy continues to shift in favor of manga.
Some of my favorite manga series:
AKIKO Story and art by Mark Crilley. Published by Sirius Entertainment. A children's story for all ages, it follows the adventures of fourth-grader Akiko as she travels to the planet Smoo to attempt to rescue Prince Froptoppit from the clutches of the evil Alia Rellapor. Beautifully illustrated in a deceptively simple, cartoony style, it perfectly captures the essence of fairy-tale adventure. I don't have kids, but people who do have told me that their kids love to have them read it aloud. I now have a special page dedicated to Akiko, with an interview I conducted with Mark Crilley, and links to all of the Akiko Reviews. Also, you can find a link to Mark's own Akiko page on The Giant Egress.

MAISON IKKOKUWritten and illustrated by Rumiko
Takahashi, english language version published by Viz Comics. A
romantic comedy by the acclaimed manga artist Rumiko Takahashi
("the Goddess of Manga"), MAISON IKKOKU is about the
slow--ever so slow--blooming of romance between Kyoko Otonashi,
the young, widowed apartment manager of the dilapidated rooming
house Maison Ikkoku, and one of her tenants, the hapless student
Yusaku Godai. Complicating matters are rivals Shun Mitaka, the
handsome, wealthy tennis coach, and Yusaku's would-be girlfriend,
Kosue, and the demented tenants of Maison Ikkoku (Mrs.
Ichinose--an incorrigible gossip, Akemi--an uninhibited bar-girl,
and Yotsuya--a businessman who's also a peeping-Tom).

RANMA 1/2 Written and
illustrated by Rumiko Takahashi, english language version
published by Viz Comics. The misadventures of Ranma Saotome, heir
to the Saotome School of Anything-Goes Martial Arts, who fell
into a cursed spring while training with his father in China. Now
whenever he gets splashed with cold water, he becomes a
pig-tailed girl (hot water changes him back). By prior
arrangement by their parents he and Akane Tendo are
engaged....could it be that when they're not bickering they
actually are falling in love? Another sort of romantic comedy
from the fevered imagination (Ranma got off lucky--other cast
members who chanced upon the Training Ground of the Thousand
Cursed Springs turn into pandas, cats, pigs, ducks, and who knows
what else) of Rumiko Takahashi.
Not manga, but manga influenced:
GOLD DIGGER Written and illustrated by Fred Perry,
published by Antarctic Press. Basically GOLD DIGGER follows the
lives, loves, and wacky adventures of two sisters (later joined
by a clone-hybrid of the two): Gina Diggers, the super-genius
would-be Indiana Jones, and her adopted sister Brittany
"Cheetah" Diggers, a were-cheetah. The thing that I
like best about this is that all the major characters are having
fun. They have weird powers and abilities, and they like
it; they think it's cool.
Some of my favorite non-manga comics:

WOLFF & BYRD: COUNSELORS OF THE MACABRE
Written and illustrated by Batton Lash, published by Exhibit A
Press. "Beware the creatures of the night...they have lawyers!"
Constantly one of the freshest and most inventive comics around.
The basic premise is simple--what if the world really was full of
all those B-movie monsters, magic, and aliens, and what would
happen when they got dragged into court for, say haunting a
house? They'd need representation, wouldn't they? Enter Alanna
Wolff and Jeff Byrd, counselors of the macabre, specialists in
the intersection between equity and ectoplasm. Batton uses this
as a jumping-off point for all sort of twisted looks at popular
culture, from the latest scandals involving super-models, to
revisiting DAMN YANKEES and Damon Runyan. Plus he has a penchant
for some of the worst puns ever committed on the English
language.
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