Most People have seen animes like Full Metal Alchemist, Bleach, Naruto, wolf's Rain and Trigun so I won't even mention them.
A movie that every anime fan should watch is X. It is by Clamp. Most people refer to it as X/1999 because it was released in 1999. It is about a boy named Kamui. He lives in Tokyo and when he was born he was granted God's Power to save humanity. There are two groups of people with psychic powers that clash. the dragon of the Earth and dragon of the Heaven. There really aren't any bad guys in this story. The Dragon of Earth want to kill all of humanity and save the Earth from pollution and destruction. And the Dragon of Heaven want to save Humans and the Earth. The Movie has alot of killing and blood so it is definitely for the more mature audience.
My second movie that I present is AKIRA. It was first a manga and it came out 1987. I know it is pretty old but the art and drawing is so accurate and detailed. It is the first anime I ever saw. It was created by Katsuhiro Otomo. It is about a boy named Tetsuo who encounters and accident with his motorcycle firends/gang called "The Capsules" why that name? no clue, but capsules are a drug that characters in the movie take. Anyways, Tetsuo goes to a government hospital and come to find out in the accident he received special psychic powers that he uses to destroy Tokyo's military and is trying to find an ancient being called AKIRA who in the past destroyed Tokyo and the explosion resulted in World War 3. Now it is in the 2030's and there is Neo-Tokyo. I could go on all day about AKIRA. It has a deep confusing background about the creation of the universe, and alternate dimensions. It is a must see and a must read!!!!
