Alright, well, I'll try not to post anything that's been posted before. I'm going to start off with feature length stuff that's more artistic and then talk about some episodic shows that I think are awesome.
1. 5 Centimeters Per Second

This is a three-part film about the life of Takaki Touno. It's a very, VERY pretty film, and is quite touching as well. I would suggest finding it in HD quality and watching it on a large screen. I don't want to say much about the plot other than it's melancholic and really good. I recommend this to ANYONE.
2. Memories

This is another three-part film, although these three have no real connection to one another. The first part, Magnetic Rose, is about a junk freighter in space that responds to a distress signal. It turns out that the signal is coming from a deserted space station. Two guys go in and try to figure out what is causing the distress signal and find haunted <3<3<3<3. The second part is more comical in tone than Magnetic Rose. "Stink Bomb" concerns a young man who shows up to work at a bioresearch facility whilst flu-ridden. At a colleague's insistence, he takes some pills. These turn out to be part of a biological weapon program—the young man soon acquires a (literally) deadly body odor and becomes a walking weapon of mass destruction. The third part, Cannon Fodder, is about a walled in society which exists primarily for the firing of cannons, ranging in size from about the size of one's hand to one that dwarves humongous buildings. One interesting thing about Cannon Fodder is that the entire thing is one continuous shot, there are no broken transitions from scenes. As a whole, Memories is very interesting. I'd suggest it if you're looking for a movie to watch.
3. Studio Ghibli Films. If you haven't seen any of these, you should. Included are Spiritied Away, Princess Mononoke, Kiki's Delivery Service, My Neighbor Totoro, Howl's Moving Castle, Grave of the Fireflies, The Cat Returns, and Nausicaa and the Valley of the Wind.
4. Akira

Somewhat dystopian future story involving mutations, lasers, and motorcycles. Interesting characters, wonderful art styles, and an intriguing, though somewhat confusing if you don't read the manga, storyline, I would also suggest this to anybody.
5. Cat Soup (Nekojiru-sou)

A very strange little story about a cat trying to find his sibling's soul in order to make her not sick anymore. Sounds strange, is strange, but is also VERY good.
Now onto series.
6. Sexy Commando Gaiden: Sugoi yo!! Masaru-san

So this is pretty much the best thing ever. It's a comedy show about a martial arts club based on a fighting style called Sexy Commando. The chief goal of the Sexy Commando is to distract your opponent to the point of leaving them wide open for attack. It's just hilarious. Trust me.
7. Sayonara Zetsubou Sensei

This is also very funny. It's about an easily depressed teacher and his homeroom class. Often a commentary on societal faults in both Japan and the world in general. Fairly lighthearted, yet intelligent humor, I think would be the best way to describe this.
8. Cromartie High School
Funny again. Nothing really intelligent about this one though. This is pretty much straight up slapstick comedy. I would recommend this for just sitting around wanting something to crack you up.
9. Azumanga Daioh

I know everyone and their mom has seen Azumanga at this point, but I would feel wrong not suggesting it. A nice slice of life that is lighthearted and funny. Based around some high school girls and their friendship with one another.
10. Onegai Teacher

The semi-inappropriate (

) story of a high school student who is forced to enter a relationship with his hot teacher from outer space. Not that he struggles much. While I admittedly am a sucker for romance anime, this was probably my favorite of that genre. I would suggest this to anyone that is a dude that is kind of a woman. If you like it, you can check out the related Onegai Twins, although it isn't nearly as good.
11. Gundam Wing

While generic, I can't help but love mechas. Gundam Wing is a mixture of nice mecha fighting along with some political strife thrown in. Don't watch this if you want nothing but action though. There's a lot of talking and things of the sort.
12. Genshiken

The Genshiken is a college club dedicated to video games, anime, and manga. Sounds kinda generic, probably, but I enjoyed this series more than 75% of the anime that I've watched. And that's quite a bit.
13. Golden Boy

Golden Boy is a jewel of a show. It mixes comedy along with a good storyline every episode in a manner that makes me watch it again and again. I like the english dubbing in this show, and would actually suggest it over subs (this is like, the only case ever.). It's the story of Kentaro Oue, a man who finished all of the requirements for a law degree from Tokyo University, but left before receiving his diploma. He roams the world, yearning to learn new things.
14. Maison Ikkoku

This is the best romantic comedy ever written. It's long, though, so if you're interested, keep in mind that it's a whopping 92 episodes with an OVA. It's the story of Godai, a student with terrible luck, and how he falls in love with a recently widowed boarding house manager, Kyoko. WATCH THIS.
15. Welcome to the NHK

The story of a hikikomori and a girl's attempt to reform him. This is a show that is sometimes funny, sometimes serious, but pretty much 100% good.
16. Vandread

A little known show that doesn't get the praise it's worthy of. It's a little cheesy, to be sure, and the 3d animation looks exceptionally old now, but it's got a good story, and certainly makes for some funny bits.
Alright, I think that I'm spent for now. Have fun. <3