There are 2 types of people, people who love and people who hate anime's. For me It is just another form of storytelling, drawn or animated makes no difference as long as the story is interesting. Watching just one anime on this list can make you a new fan of this format of entertainment. Just like in any media there can be bad ones out there, but this list will introduce some of the best ones.
1. Dragon Ball Z
Dragon Ball series is the true classic, the anime that started it all. My first large anime series I've watched, hundreds of episodes of over the top action, comedy, leveling up and beating evil. Great show for all generations.
2. Berserk
This one is as dark as it can get. World filled with dark magic and demons. Story about black swordsman and his joining with mercenary group named band of the Hawk. Kinetic action in this anime series will keep you occupied for a long time. Ending of this series is easily the darkest one I've ever seen.
3. One Piece
Good old Straw Hat Luffy. A boy who ate a magic fruit that made his body behave like a rubber. Premise is crazy but hear me out, this is an ever evolving story with thousands of episodes. Great fighting anime where everyone "levels up" and gets more and more powerful in the world full of overpowered people. His goal is to find a treasure called One Piece and to become the Pirate King. One of the longest running anime's ever.
4. Attack on Titan
This one is also as dark as Berserk. World where humanity is locked inside big walls and outside are titans who want to break in and kill everyone. Long story short Titans broke in and killed almost everyone but one brave bunch opposed them. Lots of deaths in this one, main heroes dropping like flies. Watch this one and you will not be disappointed.
5. Cowboy Bebop
Space anime about a crew of vastly different individuals trying to make some buck as bounty hunters. Music is great in this one and every character is so nicely written that it is joy to watch. Great action scenes.
6. Ghost in the Shell
Tokio Japan in near future where robots and cyborgs are reality. Everything is always involved about some mystery, it is a detective show far more interesting than say your normal CSI.
7. Fullmetal Alchemist
Steampunk story where people can transmute objects. Two brothers tried to use forbidden technique to revive their mother but that went horribly wrong and they lost parts of their bodies. Yes, sounds crazy but it gets very interesting.
8. Blassreiter
How would humans look like in our final stages of evolution? This futuristic anime answers that question. Result is overpowered creature that can level cities. Ok, I'll bite.
9. Death Note
This one is dark. A student finds strange book and finds out that writing names in that book kills that person. His counterpart is also intelligent investigator trying to find who is behind this strange deaths. One of the best written anime's out there. If you haven't seen this one i would strongly recommend it.
10. One Punch Man
This anime series is new but it has the potential to be the one of the best. It is an action packed comedy series about our hero Saitama who can destroy any foe with one punch. He is also very weird and that leads him many comedic scenes. Start with this one, you will be amused.
I'm a little watch anime, but I love them. Thank you.
I liked it very much Attack on Titan
Movies were bad though. You can read the manga for more action.
Thanks, I'll think about it.
Cowboy Bebop is my favorite, very atmospheric!
Yes, one of the best ones.
I only know Dragon Ball Z through Dragon Ball Z Abridged, which is awesome, so I have to give props to the source material.
I haven't seen Blassreiter either, but other than that I can say: great choices!