This list is like noob anime central. It's basically like saying, "Do you like jazz? I love jazz! Miles Davis is my favorite!"
(I'm not sure that I actually like anime. I just like anime auteurs like Shinichiro Watanabe, Hayao Miyazaki, Satoshi Kon, etc.)
tl;dr: Watch Cowboy Bebop/Samurai Champloo for a series and anything Miyazaki or Grave of the Fireflies for a movie.
- Cowboy Bebop (One of the greatest things ever, although the movie was meh. Also one of the only ones I can safely reocmmend in English.)
- Samurai Champloo
- Kids on the Slope (I generally don't like "slice-of-life," but I love jazz and Yoko Kanno and Shinichiro Watanabe, so I loved this.)
- Ghost in the Shell: Stand Alone Complex (A little slow at times, but very cyberpunk.)
- Most Miyazaki, but esp. Nausicaa and Spirited Away
- Grave of the Fireflies (maybe the saddest movie ever made, animated or other)
- Paprika (Took me a couple tries to love it, but it's gorgeous. I've heard it's not Satoshi Kon's best, but I haven't seen much of his yet.)
- Ghost in the Shell (I actually liked 2 better than 1, but it's been a while. Blasphemy.)
- The Tale of Princess Kaguya (super excited)
- Tokyo Godfathers
- Space Dandy
- Attack on Titan
- Full Metal Alchemist: Brotherhood
- FLCL
- Akira
- Metropolis
- Mushishi
- Paranoia Agent
- Code Geass
- Neon Genesis Evangelion
- Naruto
- Gunslinger Girl (Watched the whole first season, kept hoping for more. The concept is really good, but the execution didn't wow me. The animation took a deep dive in season 2, so I quit.)
- Hellsing
If you like sad stuff, this was sad (I guess) and had really good reviews when it aired, I watched it and thought it was just okay: http://en.wikipedia.org/wiki/Anohana:_The_Flower_We_Saw_That_Day