Keep in mind that there are lots of anime videos that do NOT have all, or even any of the qualities listed below. That means I probably don't like them. Unfortunately, the non-quality anime is getting all-too-common, with the larger(and even not so large) US video rental chains carrying more of the gorey, yeukky stuff. If you're new at this, try not to get discouraged.
That being said, here's some of the GOOD qualities of anime.
1) The art.
Yes, art. REALLY good stuff is very well drawn. Consider going to an art
gallery, and finding a picture you really like. Now imagine the picture
moving realistically. You may not consider the anime style of drawing to
your taste, but I think the above example might help you understand why
other people like it. Also, keep in mind, there are actually a few
different variations of the anime art form.
[okay, most of it is fluff :-)]
Usually, "OAV", or "movie"-length anime, has a higher caliber of artwork than a TV series. Sometimes, the same series has OAVs and TV episodes, in which case you can usually notice a difference in details.
2) The music.
Anime often has a particular style or styles of music. Particular sub-genres tend to have their own styles of music. One style is the beat-heavy almost "techno" sound, found in sci-fi anime. I find it quite agreeable. The romantic ones also have theme music that appeals to me a lot. (My favourites so far are, one or two "Video Girl AI" songs, and some "Maison Ikkoku" opening/closing songs.)
3) The silly plots.
Actually, often not so silly. Anime is far different from other animation, in that it often HAS A PLOT. And some attempt is given to add depth to the characters. Even the "fluff"-type anime (the "silly plots" I was referring to) usually attempts to get into some interesting facet of humanity. But then again, there ARE many 100% pure fluff pieces, which I find quite enjoyable ;-)
A wonderful side effect of #3 is that there is a wide variety of plot styles to choose from. There's your basic young boy-girl romantic complication comedy (ranma 1/2, KOR, ...), strange vampire/demon things (Vampire hunter D, Demon hunter Yoko,...), sci-fi stuff (robotech, Dirty Pair, Bubble-gum crisis, black magic 66), mindless violence (I don't watch many of those), and a few other things.
And they are finally putting these things in normal video stores! This is great!
Then again... not so great. Unfortunately, as mentioned above, your average video store now has
Additional tips for the first-time viewer: Don't be afraid of the subtitled ones. They are often the best way to go. Yes, there are some "good" English dubbings out there. However, there are far more quick-n-dirty ones that not only twist the words, but cut-and-splice the plot into some mindless thing that is nowhere near the original. [plus the Japanese music's usually better!]
If you've never seen a sub-titled movie, you may be amazed at how easy it is to enjoy what may initially seem to be an extremely unnatural way of viewing movies. You just have to take your pick on whether to focus mainly on the visual, or the text part of things. Depending on the video in question, the approach for best enjoyment may differ ;-)
If you are a slow reader, it may be worthwhile to watch it once looking at the words, then watch it again, looking at the animation. Sometimes the artists get a bit carried away with sneaky jokes. Especially on computer screens. I think they assume most Japanese viewers can't read English really fast, so keep an eye on when someone is looking at a screen readout that flashes a lot of information. Half the time, there will be some obscure joke hiding in seeming randomness!