My cable system recently added On Demand, which gives instant access to various movies and shows that they have on file. The movies are pay-per-view, but most of the shows are free. Even though our system doesn’t carry The Anime Network, it is offered as an On Demand option.
Now, I have to admit that my previous exposure to anime consisted solely of the shows my son watches, such as “Yu-Gi-Oh!” and the like, but I went ahead and watched a few of the shows offered, such as “Golden Boy”(What is the premise of this show? Is the title character an absolute moron or a genius in disquise??), “Najica-Blitz Tactics”(reminds me of “The Avengers”), “Colorful”(Mad Magazine crossed with the writings on the wall of a public bathroom), and the extremely insane “Excel Saga”(why the hell does Hyatt spit up so much blood, and is their emergency food supply a cat or a dog?).
What other weirdness should I look out for?
Cowboy Bebop is probably the best anime I’ve ever seen; watch it if it’s available.
Now that I think about it, most of the anime shown on Adult Swim (late-night Cartoon Network) is actually pretty good. Trigun and the aforementioned Cowboy Bebop are both great. Big O is pretty neat for a giant robot anime. Inuyasha seems like it’s targeting a slightly younger group, but it’s still fairly popular and reasonably entertaining. FLCL is kinda trippy; I can’t really comment on the series as a whole (6 episodes, I think) because I haven’t watched it all in one sitting and don’t know the order of the episodes. They play some other anime as well, but it either never attracted my attention or simply isn’t as good, in my opinion.
Menchi, Excel and Hyatt’s emergency food supply/pet is, in fact, a dog. I know it looks an awful lot like a cat, but listen for the dog-like noises.
Hyatt coughs up blood because she, well, dies a lot. Sometimes it’s not very clear in the anime, but the original manga makes it a bit more explicit. The manga’s crazy, too, but Shinichi Watanabe (you know, Nabeshin) just went completely off the wall with the anime. By the way, Shinichi Watanabe really does look like that, afro and all.
I don’t really know about the Anime Network, but The Ping-Pong Club makes South Park look like Hey, Arnold! at times. If you get a chance, you should watch My Neighbor Totoro, Spirited Away or Irresponsible Captain Tylor.
It’s a parody series. Hyatt is meant to be a type commonly found in anime – the sickly waif. If you watch enough anime, you’ll see dozens of them: sweet, sensitive girls with serious health problems, usually heart trouble.
I’m not sure what’s on the Anime Network, but I’ll throw out some general recommendations.
If weird’s what you’re looking for, I recommend Evangelion (Hideakki Anno’s reflections on Carl Jung, done as a giant robot anime), and Serial Experiments Lain (which is all about the Internet, God, and the nature of reality). I’ll second the recommendation of FLCL, but since it’s a parody series (mostly of Evangelion and Anno’s other major work, His and Hers Circumstances), it’s hard to get the jokes if you see it first. Cowboy Bebop isn’t very weird, but it’s definitely good.
Revolutionary Girl Utena is also wonderfully strange.
I second watching out for Revolutionary Girl Utena. That has to be my favorite anime series. The movie’s animation is just fantastic (but completely a mystery if you haven’t watched the series).
I haven’t really kept up with anime for the past five years or so, but I do recommend Record of the Lodoss War (fantasy) or it’s sequel series, Sakura Wars, Martian Successor Nadesico, Robot Carnival (movie of shorts), Blue Seed, and Haibane Renmei (which is newer, but a very quiet, peaceful sort of anime).
Looking at what AN offers on their website, if you want comedy I found Those Who Hunt Elves to be pretty amusing, and I liked the manga for Sorcerer Hunters, but know nothing of the anime (I heard they sort of… er… made it “risque”).
My opinions are probably very out of date, but I hope this helps a little.
Do they have any Hentai Anime 
Because the Anime Network is run by ADV, and I believe that they are only running shows ADV has the US rights to, I would say no.
Does this network show uncut anime, or is it edited according to US TV standards (no nudity unless you’re Detective Sipowicz)?