I'm surprised how feeble most of these lists are:
Cowboy Bebop (I'm not a fan, and the movie isn't essential, but it's a classic and most people enjoy it).
Full Metal Alchemist (the movie isn't essential, but most would find it enjoyable).
Ghost in the Shell (Ghost in the Shell, GITS 2: Innocence, GITS: Stand Alone Complex and GIT:SAC 2nd gig are essential, but Solid State Society and the two summary films aren't, though most would probably find the former worthwhile, and some the latter).
Grave of the Fireflies.
Neon Genesis Evangelion (and NGE: The End of Evangelion).
Princess Mononoke and Spirited Away.
Voices of a Distant Star.
Wolf's Rain (a personal recommendation and a good fantasy series).
After watching all of those, I doubt anyone could insist anime has nothing to offer.
Also, I haven't seen Texnolyze, Noein, Monster, Haibane Renmei or Deathnote, but they might all be worth looking into, and I haven't finished The Melancholy of Haruhi Suzumiya (or started Lucky Star), which I imagine are popular, but not essential series, from what I've seen so far.
Vampire Hunter D: Bloodlust is an excellent showcase, particularly the rise during Carmilla's ressurection scenes, up until the end of the fight between D and Link. Azumanga Daioh, Fruits Basket and Pokemon are also recommendable, particularly for an understanding of classic anime and Japanese culture, but aren't essential (especially Dragonball and Pokemon in their entirety). Naruto's Haku saga stands out as the highpoint of the series, and is worth watching, even isolated from the rest, while shounen make a great case study, including Bastard, Bleach, Dragonball, Naruto and One Piece. Inuyasha and Rurouni Kenshin are good shojo equivalents, where Loveless is also a highly recommendable modern series. Some might call different series in this list essential.
And, of course, I will have missed some deserving films and series.