I guess I'm in the same batch as Jett, what with having watched anime since I was pretty young and ended up developing a varied interest. I'd probably blame that on my dad, who held - and
maybe still holds - an interest in the mecha and shounen anime genres. Got me hooked on one of the Gundam series when I was in kindergarten, which eventually snowballed and got me into anime in general. Unfortunately I never did get back into Gundam since I'm too lazy to watch a bazillion episodes and whatnot. But never mind that business, time to answer some questions.
In terms of favorite anime, I should probably mention I have different categories of favorites. For example, I might have an anime whose world is one of my favorites but I hate the story/characters, and vice versa. Though taking into account performance in every parameter, the first few that come to mind are A Certain Scientific Railgun, Fate/Zero, Bakuman, Katekyo Hitman Reborn!, and Yu Yu Hakusho in no particular order. Honorable mentions go to Code Geass, Tengen Toppa Gurren Lagann, Durarara, Dragon Ball Z, and Death Note - all of which I've watched so long ago I'm not sure whether or not they'd be in the list. Although I'm fairly confident that 2-4/5 would manage to get in. There's likely a few more I'm missing that I like, but since I can't remember them I guess they aren't really favorites
As for characters, a few stand out. Rider and Kariya Matou from Fate/Zero, Kamina from Gurren Lagann, Shizuo Heiwajima and Shinra Kishitani from Durarara, Kurama from Yu Yu Hakusho, Armin Arlert from Attack on Titan, and Tsunayoshi Sawada from Katekyo Hitman Reborn. Honorable mentions are Accelerator from A Certain Scientific Railgun (for his performance in the light novels and the second season of the sister series), Moritaka Mashiro and Akito Takagi from Bakuman (for their character dynamic), and Keima Katsuragi from The World God Only Knows.
I can't really say much about what I like about anime since Jorick's got a pretty good handle on basically everything I would probably say on the subject. Though what I
will say is that I notice that anime tends to have better music - soundtracks, openings, endings, what have you - than most of the live-action or cartoon series I've watched. I will admit that's not a whole lot of series, however.
I don't think I could confidently recommend an anime to the world, what with everyone having their own personal tastes. But if I absolutely have to, I'd definitely suggest people take a gander at Fate/Zero. Unfortunately, there's a few glaring flaws I should point out. The first episode is basically an hour of infodumping, there's one or two filler episodes that seem to barely serve a purpose, and, due to it being a prequel, has to deal with (what I think is) a rather anticlimactic ending to set up Fate/stay night. But apart from those flaws, I'd call it a pretty balanced out anime. It's only 25 episodes, which you don't have to use a few days worth of time to watch it like Naruto or something; the writing of both story and characters is spectacular; and the fight scenes are impressive as hell, especially since the animation budget makes some parts look nearly cinematic in quality.