I'll just lost my typical Anime for people who aren't yet fans of anime list:
Anything by Shinichiro Watanabe, this includes Cowboy Bebop, Samurai Champloo, Kids on the Slope, Terror in Resonance, and Space Dandy (this one is better for people who already really love anime). He's my favorite anime director and one of the best at creating works that bridge the gap between Japanese and western audiences. His shows have a big focus on music, which is nice if you are a music buff (especially the shows that feature Yoko Kanno; she's incredible)
Full Metal Alchemist (Brotherhood): Has a very Western-inspired design for the world and characters, no complex Japanese names or odd cultural things that a western audience wouldn't understand. The dub is amazing, and while in some respects this show can be a typical shonen series, it is much more mature and complex than most shonen-esque series. Great narrative, great characters, and an interesting setting with struggles and parallels to our own, especially in the west. A must watch.
Psycho Pass: Apparently some people are down on this show in this thread, but imo it is nothing short of fantastic. Incredibly well-paced, top-notch writing, with a blade runner meets 1984 style setting that is a very compelling backdrop for what basically amounts to a detective thriller. Very similar to western police/serial killer dramas, and the characters and their relationships are quite compelling. Also features one of my favorite anime villains. Definitely a must watch.
These are the ones I would recommend right away, but there are plenty of great series out there for anime newcomers.
Edit: Also Mushishi, a very approachable and interesting supernatural themed anime. Has a good dub as well.