This list is fairly short, because although there are hundreds of recent anime that aren't bad and quite watchable, they are mostly forgotten two years after release. It's hard to think of many that rank as outstanding titles that those new to, or curious about, Japanese animation should see. With comments to aid those new to Japanese animation (anime).
FLCL: Bizarre events involving a junior school boy, his older brother's girlfriend, and a weird girl with a bass guitar and a motor-scooter. And some robots.
Read or Die: The 3-episode OVA version. Yomiko Readman is both a book geek and an intrepid secret agent, in a fast-moving alternate-world action adventure.
Earth Girl Arjuna: SF with strong ecological message. Girl develops strange powers in response to ecological crisis. Great visuals.
Niea_7: Girl student flunks college and has annoying alien girl billeted in her room. Quietly and blackly comic.
Serial Experiments Lain: Cyber weirdness involving introverted and withdrawn schoolgirl.
Ghost in the Shell: #2 -Innocence: High-budget SF movie, best seen in cinema, about crime in a world where almost everybody is a cyborg.
Appleseed: (new movie). High-budget movie, not a bad story, but more impressive for its cutting-edge computer animation than anything else.
Twelve Countries: 45-episode fantasy adventure about a schoolgirl who is carried off to another world where she exerts magical powers. Familiar plot, but creates its world in unprecedented detail, and the heroine develops as a person.
Spirited Away: Another hit from the renowned Japanese Studio Ghibli. Small girl stumbles on a town where strange beings gather after dark, and her parents are trapped there.
Howl's Moving Castle: Recent hit from Studio Ghibli, and adapted from a novel by Diana Wynne Jones, which accounts for its strongly European flavour.
Perfect Blue: Pop singer is pressurised by her management to quit her 3-girl group and take a part in a TV series. Stressed by the TV shoot, disgruntled fans and a violent stalker, she loses her grip on reality.
Scrapped Princess: Fantasy story about an unwanted princess that eventually turns out to be something else altogether. Love the opening theme music.
Cowboy Bebop: Punk SF story in which two bounty hunters and a girl scratch about the solar system in an old space-ship, trying to make money and forget their pasts.Most Japanese animation has child or adolescent principal characters (in common with its principal target audience). At one time, it was usual for the lead character to be accompanied by some cute animal. If this sort of thing irks you, I can suggest some alternatives:
1): Material full of war or gritty violence, police procedures, and the like, which naturally has adult characters. Such anime is not particularly well represented these days, but there were a lot of OVAs made which qualify:
Ninja Scroll (and lots of other ninja stuff)
Dagger of Kamui
TV series? I can only think of Zipang, Hellsing, Ghost in the Shell SAC, and the lengthily titled 'Kochira katsushika ku kameari kouenmae hashutsujo' (an obscure police comedy series), Iresponsible Captain Tylor, and Patlabor (TV). Sophisticated? Sometimes, but this is also the kind of material that violent-minded adolescent males particularly enjoy.
I'd define this as anime in which the principal characters are young adults engaged in the kind of things young adults commonly are preoccupied with, and the tone of the anime suggests that it is aimed at this same age-group. This kind of anime - usually a TV series - has become much more noticeable in recent years.
Honey and Clover
Kimi ga Nozomo Eien
etc. You know you are watching a young-adult anime when two of the characters clearly have sex in the first episode. These anime are generally concerned with romance and relationships, which topics will be of interest to many, and these anime can be quite sophisticated.
There are still more anime, still with adolescent characters, which are made with more than usual sophistication:
Zettai Shounen: Slow-paced and thoughtful revelation of secret of mysterious magical flying objects
His and Her Circumstances: Masterful (and funny) study of teenage self-absorption among clever youngsters.
Finally a few more suggestions, which I have included purely because they were fun.
Wife is a magical girl: You need to know what a magical-girl anime is usually like to appreciate why this is so funny. The heroine is a grown woman who has trouble fitting into her skimpy magical-girl outfit, she has a teenage magical rival, a husband, and a boyfriend.
Love Hina: A rather average schoolboy finds lodgings at his aunt's all-female lodging house. The place is full of female students of varying ages, and any lechery from him, real or suspected is, usually, firmly slapped down by assertive young women.