I have a lot of things that could fit into the favorites category if I took the time to think it through. The ones that spring immediately to mind are Monster, Fullmetal Alchemist: Brotherhood, Code Geass, Tengen Toppa Gurren Lagann, and Bakemonogatari. I'm also thinking that Kill la Kill will be up there in the favorites list, but I like to reserve final judgment until I watch the entirety of a series and that one is still airing. There are definitely a ton more I could list that I like as much as the ones I've already mentioned, but this will suffice.

What do I like about anime? Eh, I dunno, same things I like about other kinds of entertainment media. I see a lot of people talk about anime like it's some alien form of media that has only the slimmest of commonalities with other forms of fiction, but I've never seen it that way myself. They're all vehicles to tell stories, whether it's animated or live action or a book or spoken word or whatever. The thing I like about animated things (anime included) over other forms of fiction is that they can show all sorts of crazy things without breaking internal consistency and thus immersion. A lot of special effects things in movies just look fake as hell and the clash between the two levels of fiction, the fiction of the people doing the things you're watching and then the fiction of the monsters or explosions or whatever that don't look real, forcibly jars the mind and reminds you that it's all fake. In the realm of animation all you have to do is maintain a consistent art style and anything you throw in will look like it belongs, so even though you're seeing something patently absurd that could never exist in reality you can still be immersed in the fictional world being presented to you without ever being like "wow, that looks so dumb and fake." Movies are getting better about that as technology advances, but animation just has a huge leg up on it due to the nature of the medium. I suppose this internal consistency also allows anime makers to do things farther outside the box than live action media could ever even attempt, and I do enjoy crazy bullshit going on in things I watch (just take a look at the favorites I listed, Monster is the only one that doesn't have over the top things going on), so I guess that's another reason why I like anime.

Favorite characters, hmmm... Wolfgang Grimmer from Monster. Roy Mustang, Lin Yao, Scar, and King Bradley from Fullmetal Alchemist. Lelouch Lamperouge and Jeremiah Gottwald (the hammiest son of a bitch ever, gotta love him) from Code Geass. Kamina and Simon (more so in the second half of the show than the first) from Gurren Lagann. Hitagi Senjogahara from Bakemonogatari. Pretty much all of the major characters from Kill la Kill are great for various reasons. I could go on listing more, but I figure just listing my favorites from the anime I listed as favorites is good enough.

If I were only allowed one recommendation and it was to the world at large, no qualifications or whatever based on preferences or past experience with anime, I'd have to go with Monster. It's not crazy action stuff, the art style isn't very stereotypically anime, and it's an amazing story, so all that combined makes it the best recommendation I can think of for general use. Even for people who are well versed in anime and are fine with nonsense and anime art styles, the story is so damned good that it's worth the recommendation. It's easily the most solidly crafted story I've ever seen in anime, and it competes with live action TV there too; if I were to liken it to live action TV shows, I'd rank it up there with things like Breaking Bad and The Wire for pure quality storytelling, and those two are ones I list up in my favorite shows of all time. I'm probably over-hyping Monster at this point, so I'll just stop now. :lol