Right, I was hoping to see an anime thread pop up. I was giving this some thought recently, and I came up with some things I think need to be addressed when you're going to recommend some anime, especially to a first-time viewer.
There are three things that a recommendation needs to avoid:
1) It cannot have an ultra-thick, ultra-confusing plot. Some anime go really deep into things - to an even greater extent than movies a lot of the time - and if the person isn't ready, they can feel overwhelmed. Neon Genesis Evangelion is a prime example of this, especially since if you try and treat it as a whiz-bang robot smasher, you're going to be disappointing.
2) It cannot be too steeped in Japanese culture. Having to stop the video every few minutes to read translator notes about some Japanese custom does not make a good viewing experience (or worse yet, watching the dub and not even having the chance to read an explanation). Something like Welcome to the N.H.K. or Ouran Highschool Host Club fit this description.
3) It has to be in a genre that the person asking for the recommendation likes. Some shows are aimed at specific groups of people with specific tastes. School Days is really good at what it does, but what it does only appeals to certain people (those being sadists and fans of classical tragedy).
So, what would I recommend?
- Full Metal Panic. While it's certainly not one of my favourites (not by a long shot, actually), it's very competent and well-executed. It introduces the idea that a show can feature giant robots, but not actually be about giant robots. Also, it has a nice sprinkling of Japanese culture to get a viewer familiarized with that. Code Geass is in the same boat but instead of introducing Japanese culture, it can help a new fan ease into the idea of having to sit down and think about how the supernatural events in the show are working (as opposed to just being handed all the information), as well as having to keep track of a massive cast.
- School Rumble. While it does seem to go against point 2 above, what with some of the really, really Japanese things going on, it still doesn't feel alienating, and much of what goes on is highly relatable to teenagers of any country. The Melancholy of Haruhi Suzumiya could do this as well, so long as the person is aware of the Endless Eight and has some sort of plan for that.
- Full Metal Alchemist: Brotherhood. I haven't seen the original, so I don't know if that might be a better choice, but this one is a very good take on the 'superpowered teenager' story, as well as having a large, colourful cast to keep track of. And unlike things like Bleach or One Piece, this only has 64 episodes, so you won't be stuck for months trying to slog through to the latest end of the other two.
- Baccano! and Black Lagoon. Both are good examples of shows that pay little heed to any of the cliches someone might expect from anime. If the person is old enough to handle the violence and mature themes, then they will probably enjoy these, though Baccano!'s confusing plot may be offputting (see point number 1 above)
- Trigun and Cowboy Bebop. Personal feelings about these shows aside, they both have a lot of mass appeal, so I can't very well make a post full of recommendations without including them. Of course, this means I can't give justifications other than 'mass appeal', but that's what other people are for, right?