This is a general list of "must-watch" anime oriented for those who are new to the medium. Each title that is added to this list should have at least a small description on its own page. The following shows are popular and critically acclaimed, though some may not align with your own personal taste.
Also check out EcchiMaster's Recommendations (nsfw)(if you're into that kind of thing).
For other general recommendation charts check the /a/ wiki . In fact this wiki is loosely based on this chart .
Editor's Note: Please keep each genre limited to 12 recommendations each. Any more than that has a tendency to be overwhelming to people new to anime.
Action
Comedy
Hayate the Combat Butler!
Drama
Anohana: The Flower We Saw That Day
Kimi ga Nozomu Eien (Rumbling Hearts)
Fanservice
Fantasy/Adventure
Harem
Horror
Magical Girl
Mahou Shoujo Lyrical Nanoha
Martial Arts
History's Strongest Disciple Kenichi
Mecha
Mobile Suit Gundam Movie Trilogy
Macross: Do You Remember Love
Movies
The Girl Who Leapt Through Time
Music
Mystery
OVAs/ONAs
Rurouni Kenshin: Trust & Betrayal
Psychological
Romance
Ef - A Tale of Memories & Ef - A Tale of Melodies
Sci-Fi
Legend of the Galactic Heroes
Now and Then, Here and There
Slice of Life
Croisée in a Foreign Labyrinth
Tamayura & Tamayura: Hitotose
Sports/Games
Supernatural
Honorable Mention/Anime That Didn't Fit in Other Genre