Everyone above laid out some good points, recommendations, and information on anime and manga, as well as the appeal of it. I figured I would take a little crack here, myself, though.
Basically manga are Japanese comic books. But unlike American comic books, they usually are not released every month and they are usually black and white, with colored pages being used on special occasions. Manga typically come out once a week and can number from as little as 9 pages up towards 22 or so pages per chapter. They are put together by a team, usually. For the most part the same creator does the writing and illustrating, with help from assistants for backgrounds, inking etc... Sometimes an artist and writer team up on a series, this is not the norm, but it seems to be happening more and more lately given the popularity of light novels in recent years. A lot of those writers are trying their hand at manga writing and they are teaming up with artists.
Manga are usually published as serials in manga magazines, these look like huge phone books, and the magazines come out weekly, monthly, bimonthly, etc.. They are pretty affordable given the amount of content within them. As others have mentioned, getting into these magazines is both very tough, and once you are in it is very competitive to keep your series published. The magazines run reader polls constantly and readers send in their votes, if a new manga doesn't perform it is on a very short leash. For every super hit like Dragonball, One Piece, Naruto, and Bleach, there are countless failures that may have only received one tankobon (a digest sized collection, what we would call a trade paperback, but digest sized). All of those books you see in B&N are localized tankobons. They collect the serials from the phone books into digest sized trades.
If a manga is popular enough, it will usually receive an anime adaptation. Given the length popular manga series tend to run it is not uncommon for the anime adaptation to catch up with the manga story-line, then the anime staff have to use tricks such as over extending shots (remember how everyone always screamed for a long time when powering up in Dragonball Z?), lengthening the opening and ending songs used in the episodes, adding extra content to the story-lines they are adapting, and sometimes even doing entire original arcs, called filler arcs, to give the manga time to get a head again. There is big money at stake once the series becomes an anime, the merchandising alone is crazy. Mangaka can become multimillionaires with a hit series.
As others have mentioned, manga supercede super hero comics, there are so many genres of manga it would be crazy to try and name them all.
I should also note, that anime do not need to be adapted from manga. Anime are adapted from manga, light novels, video games, and basically every other medium. There are also original anime (which I tend to prefer as I enjoy reading the original manga over watching an anime adaptation).
My personal recommendations for members of this board, most of these are available in one way or another, and it's probably safe to say these are for 16 and up.
Manga with anime adaptations, remember when in doubt, read the manga.
Berserk – This is an insanely dark fantasy comic. Definitely for adults only. Watch the original series from the 1990s or better yet read the manga. All other animated adaptations have funky art due to being modern.
Opening:
Bastard!! - Again, dark fantasy, adults only. I would either read the manga or watch the Netflix adaptation.
Netflix opening:
Hajime no Ippo – A boxing anime and manga. The manga hasn't been licensed but fanscans are out there. If not, the anime is pretty damn good, and really funny at times. Great fights and slice of life stuff.
First opening:
HunterxHunter – This is a Shonen (aimed at young men) manga like Naruto, DBZ, One Piece, etc... but is really a Seinen (something geared toward older men) in disguise as the story progresses. Read the manga or watch the anime, but the manga is better.
Original anime:
Cowboy Bebop: A masterpiece blending blues and science fiction. Great music.
Opening, a masterpiece:
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Neon Genesis Evangelion: A masterpiece that deconstructs the mecha genre. Watch the tv series, then End of Evangelion, then the 4 movies.
Opening, one of the greatest of all time:
Anime based on a light novel:
The Rising of the Shield Hero: This is an isekai, ala John Carter of Mars. A guy is sent to another world where he becomes the Shield Hero. He's falsely accused of rape and is considered an outcast by everyone, including the other three heroes summoned. He has to work his way back up from the bottom and becomes a force to be reckoned with.
Opening Theme of Season 1:
