Anyone who's a fan of anime and has a Wii needs to get Fragile Dreams: Farewell Ruins of the Moon. It's a ridiculously beautiful game, both in terms of strong atmosphere and a somehow tragic but hopeful story. But it's also a bit melancholic; I could only play through it once, because the entire game is just so emotional. The American production team, Xseed, also did a fantastic job with the voiceovers. I honestly can't think of anything bad to say about this game. But I feel like it'd be right up the alley of a casual gamer who likes anime.
On to less detailed and sappy descriptions:
-On the DS, I'd recommend the Etrian Odyssey series. They're tons of fun, but they might be off-putting to casual gamers - the earliest levels are the hardest. To fans of strategy or adventure games, though, it'd be awesome.
-On the PC, I can only think of Vampire the Masquerade: Bloodlines. It's old, and it's glitchy, but the voice acting is great, the graphics are good for its time, and there are still fans creating independent patches for it. The gameplay isn't perfect, but the whole thing is still a ton of fun. This one has a darker, more adult feel than most games, though. It's not the sort of game everyone will like, but fans of supernatural/gothic/punk-type things will have a blast.
Mm, other than that, casual gamers should try the Harvest Moon or Rune Factory series. Both have heavy anime influences, and (for the most part) won't lead to any rage-quitting.
But seriously, check out Fragile Dreams. It's so...so...
lovely.
....I feel like I've just taken this all way too seriously...