You need to tell them the difference! Cartoons are one-offs with everyplots and zero continuity. Anime is the exact opposite of that!
Sorry to be so anal about this but "Anime" as defined, is simply Japanese animation. Much of it is as shallow and as poorly executed as the generic tripe we've grown up with in the west.
Ooo, I almost forgot. Before I go to bed, someone here HAS to tell me they're a fan of Tengen Toppa Gurren Lagann. Please. You don't know manly until you've seen that shit.
Orange shades = manliness cubed
I know I'll be repeating what some have written above but some, I feel essential viewing for ye olde burgeoning anime fan includes:
Cowboy Bebop, arguably the the best all-round series out there and undoubtedly the best for converting the uninitiated or skeptics into fans. Unless you're under fifteen or utterly abhor action, it cannot come more recommended.
Death Note, a masterfully executed exercise in suspense that raises some genuinely troubling and ambivalent emotions in the viewer.
Akira, which perhaps only along with
Spirited Away (also recommended, along with any other Studio Ghibli titles) is an anime that's achieved mainstream acceptance.
The Vision of Escaflowne, has aged visually and has somewhat left of centre character designs, but is
still visually arresting, with outstanding plotting, direction, voice acting (in Japanase, the English dub is a mixed bag) and character interaction, and was a unique fusion of shonen and shoujo that blended together seamlessly.
The Irresponsible Captain Tylor, is as fresh and appealing now as it was in the early nineties and is the perfect antidote to anyone who finds they can't turn on a television these days without discovering a clumsy, sanctimonious sci-fi show that takes itself way too seriously, on every other channel.
Berserk, a drama-horror set against the back-drop of some of the best realised historical fiction this side of Guy Gavriel Kay. Be warned; it makes holocaust movies seem light-hearted.
Once you've these under your belt, I might recommend
Neon Genesis Evangelion (if only to understand what all the fuss is about),
Fruits Basket (for its heart warming/wrenching cast),
Ghost in the Shell and
Ghost in the Shell: SAC (which make most live-action sci-fi series/movies look ham-handed and awkward by comparison) and once you've digested a
significant amount of anime, have a gander at
Excel Saga.
And if you like to get the old gray matter stirring, watch and be sucked in by
Serial Experiments Lain.