I'll skip the movies since some great ones have already been mentioned, and just list some series from different categories that I liked. Might as well put a brief summary of the plot, so sorry if it's a bit lengthy.
Action -
One Piece (gotta watch the Japanese voice, english sub tho, English voice version butched this gem). One of the best anime series, IMO. Pirates, adventure, friendship theme. Really tight and good story arcs tho, fighting is top notch, and everything makes sense within the rules of the OP world.
Bleach. High school guy gains powers of a death God (shinigami), fights creatures called Hollows who are spirits who get twisted by resentment, hate, ect. Bigger plot unfolds making him visit the death God society to save a friend.
Naruto. Kid is born and has a tailed demon put in him. He grows up with everyone hating him, causing him to be obnoxious to get attention. Fighting deals with ninja techniques, and follows a mission structure early on as Naruto lives in a town where ninjas are rented out for the right price.
Katekyo Hitman Reborn!. Tsuna is a kid who everyone calls a loser. Too afraid to do anything, and an overall scardy cat who sucks at school work, athletics, everything. A hitman tutor who looks like an infant named Reborn shows up, telling him he's the next Vongola mafia boss, and that he's going to train him to become it whether Tsuna wants to be mafia or not. Loved the series personally, but the first 30 episodes are a bit slow, but after those...next 173 are great heh.
Full Metal Alchemist Brotherhood (same as Full Metal Alchemist, but follows the manga closer, so the story deviates and is better IMO). Deals with 2 children who live in a world where transmutation can be done by drawing on surfaces and putting the components within it. They try to bring someone back from the dead, lose body parts in the process, and set out to find a way to become whole again.
Kenshin. World's greatest swordsman retires, and refuses to kill. He's gradually drawn back in. Great supporting cast, and a good story overall. I liked this one a lot too.
Code Geass (action with a touch of Death Note in it). Lelouche gains the power to make anyone do anything he says one time. Uses it to get revenge and play with a war like it was a chess game. Has mechs as well.
Cowboy Bebop. Just an overall fun series, a bit short tho. Every episode it pretty self contained, and entertaining. The English voice work I hear is better on this one (never watched the Japanese version myself). Basically it's a bunch of bounty hunters traveling around trying to collect bounties. The artwork is great and fluid as well.
Ghost in the Shell, stand alone complex. A prequel series to the movie. waayyy too much going on to summarize in a small paragraph lol.
Gundam. By that I mean there's lots of them, and don't want to give each it's own paragraph lol. My faves are the original 0079 series, Gundam Zeta, Gundam 08th, Turn A Gundam, Gundam 00, Gundam Wing, and G Gundam. All of them are more war like with drama and action except Turn A and G. Turn A still has war, but is much more light hearted (which is actually a real benefit to the series), and G is more over the top super robot fighting.
Yu yu Hakusho. Tough guy school kid dies and becomes a spirit detective. Series is really good, another one of the classic much watch series, again IMO. Lots of good fighting, and the stories not so bad either.
HunterXHunter. Pretty sure the manga for this was written by the same guy who made Yu yu Hakusho. The anime is good, but cuts off before the ant arc. It has action, but in a slightly different way until about mid way thru the anime. Of course then again the ant arc in the manga is such a high point, low point arc...maybe it's better if the anime didn't show it just yet lol. Seriously, never has a manga arc litterally from week to week had me going, "Awesome can't wait for next week.....what the **** is this!?!? This is crap....ok things are awesome again....what wait 1 year for the next manga issue!?!?", like this series has lol. Still tho, anime never got to hit that trouble point lol.
Drama/Serious/Mind **** -
Death Note. Light gains a book, where any name he rights in it causes the person to die. He decides to try and change the world the way he wants it. L is a super genius who tries to figure out who "Kira (Light's killing alter ego) is. Major selling point of this anime is the clever back and forth mind games inbetween Light and L.
Elfen Lied. A group of women who aren't human, and are able to take completely dismantle a body with invisible hand like apendages. One gets lose, a guy tries to befriend her, death follows suite lol.
Ergo Proxy (tho it fits in the action category a bit too). Future apocolyptic looking world where there seems to be only one city. A goth looking woman named Lil Mayer (think it's Lil...or R'eal or something like that...been a bit since I saw it) is partnered with a robot and looking at the scene of a crime. She sees a creature called a Proxy that kinda turns her world upside down. Not sure how much more I can say on it lol, It's a good one tho.
Action-Comedy -
Toppa Tengen Gurren Laggan. Over the top and pure awesome. If you like awesome characters, action so over the top it becomes comical, great catch phrases, and mechs, this is a much watch IMO.
Trigun. Vash becomes the 1 million dolloar man, bounty wise. Everyone wants to catch him, and he just wants to not get caught. 2 insurance agency ladies follow him to try and stop him, as well as represent their insurance agency as Vash is know as the human stampede (everywhere he goes accidently ends up demolished). Story slowly branches out, overall a very fun anime to watch tho.
FLCL. Coming of age, over the top anime. Made by the same ppl who made Toppa Tengen, with the same awesome over the top feeling. Definitely worth a watch as well. Pretty short series like Cowboy Bebop tho.
Comedy-Romance -
Tenchi. Normal guy gets a harem of crazy women, meanwhile he just wants them all to leave so he can live a normal life. Typical harem anime set up, but this one kinda paved the road for a lot of them that came after it.
Clannad . Odd one to place. First part is more comedy and romance. Last half fits more in the drama/serious section. Guy with tough attitude falls for innocent sickly girl. He gets a bit of a harem going and it has decent comedy. Second have just goes all depressing, completely different show at that point, but still good.
There's more, but can't think of them atm. Anyway hope those help a bit.
