Kekkaishi
An action series about two childhood friends who fight off monsters called "ayakashi" that attack their school grounds every night as it sits on an area that oozes immense power. It contains good role models and avoids a lot of clichés. The female lead, Tokine, is also handled very respectfully. Very high recommendation. It's got good storytelling, good characterization, and good action scenes that rely on strategizing, rather than competing to see who can scream the loudest.
Astro Boy (2003)
A fairly psychological children's sci-fi action series about a robot who comes into a world filled with humans who have a prejudice against robots and vice-versa.
The flower we saw that day (AKA, "AnoHana")
A bunch of childhood friends began to drift apart after one of their own died in an accident. Years later, she appears to one of them and he's the only one who can see her. They try to bring everyone back together. The narrative strives to put you in tears.
BoBoBo-Bo Bo-BoBo
A man with a blonde afro uses his nose hairs to battle the forces of evil after Czar Baldy Bald IV takes over the world and decrees that all of mankind becomes bald. In case you couldn't tell already, this is a very ridiculous series. It's an action-packed parody on animé itself and beneath all the layers of madness lies a work of genius.
Fullmetal Alchemist Brotherhood
After an alchemical accident, two brothers ended up losing a lot. The older brother lost an arm and a leg, while the younger lost his entire body, leaving his own soul bound to a suit of armor. The older brother was given metal prosthetics and a spot on the military, where the two of them have access to extensive research in how they might be able to reverse the effect and return their bodies back to normal. The story takes place in the early 1900s.
Fooly Cooly
It's weird, it's short, and it's about growing up. It also involves someone in a bunny costume using a flying guitar to battle a giant monster. Also, random stuff keeps literally popping out of the main character's head, including a fully-functional robot.
Usagi Drop (no Dub)
An old man had an illegitimate child and has just died, and the other family members are just trying to avoid her and want her out of their lives. So her adult nephew decides on the spot to adopt her and soon begins to realize what a gargantuan change that fatherhood has just placed on his life.
Durarara!!
The focus shifts between loads of different characters and their lives in a city. Sometimes, it's pretty normal, like some guy bringing people together to do voluntary community service, while other times, it involves a supernatural woman searching for her missing head.
Little Busters
A group of childhood friends is in Highschool and they will be graduating soon and they will inevitably go their separate ways. So they decide to let their youth go out on a high note with a game of baseball. But they themselves only make up five people. So they need more players and start to look for some new younger friends to join them. The show itself is not actually a sports animé. The whole "baseball" aspects plays only a very minor role.
Is this a Zombie?
The protagonist is murdered by a serial killer. However, this one girl whom he had cheered up earlier that day turned out to be a necromancer and resurrected him as a zombie. He wants to uncover who his killer was. A magical girl and a vampire ninja join him on his adventure. Wait, what!?
My Bride is a Mermaid
Some guy winds up getting himself involved with a mermaid mafia and has to accept an arranged marriage with the mob boss' daughter one day or be killed. He gradually begins to like her. This series places a strong emphasis on the fact that marriage doesn't just bind two people to one another, but brings two entire families together. It isn't laser-focused on the two lovers. By the end of it all, they share the same circle of friends and it brings attention to still keeping other people in your life.
---THE FOLLOWING CONTAIN FEMALE PROTAGONISTS---
Squid Girl (only the first season has been dubbed)
A humanoid squid-like entity holds the desire to conquer all of mankind and attempts to do so on her own. She gets as far as breaking a hole in a restaurant and has to pay off the damages. Hilarity ensues.
Fruits Basket
A girl gets herself involved with a family with a curse that causes them to be turned into animals when they're hugged.
Joshiraku (no dub)
Each episode involves a bunch of friends discussing something, but gradually the conversation gets more and more way off-topic and things escalate like crazy.
Kobato (no dub)
A woman is partnered up with a talking plush dog to solve people's issues or ease their pain and trauma.
Lucky Star
Pretty much the quintessential slice of life comedy.
Tokyo Mew Mew (dub was never finished... and said dub was by 4Kids)
Magical girls with animal-based traits (yet are named after food for some reason) battle mutant behemoths lead by aliens that want to take over the world.
Dream-Eater Merry
A resident of the dream world has gotten herself stuck in the human realm and wants to return home. She likes doughnuts.