Probably going to repeat a few series that have already been recommended and I don't know what you will like so I am going to just chuck a few with a quick description.
Angel Beats!: Guy wakes up dead and finds himself in pre-heaven. It's got humor and emotions in it. It's hit and miss and they did rush it a bit.
Azumanga Daioh: My absolute favourite anime ever. It's short sketches about the lives of school girls. It doesn't follow much of a story.
Chuunibyou demo Koi ga Shitai!: Airing at the moment. Has some very good fight scenes and is very entertaining. Probably going to be a typical anime end though, which always disappoints me.
Clannad: If you feel like being sad, then this is a good show to watch. Clannad: After Story is more emotional, but you need to watch this first to really bond with the characters.
Danshi Koukousei no Nichijou: A.K.A The Daily Lives of High School Boys. It's another slice of life comedy but with boys this time.
Durarara!!: Very complex and requres a bit of reading, but it has very good character development and story along with some action.
Gyakkyou Burai Kaiji: Ultimate Survivor: Nothing happens, but IT'S SO INTENSE. It has cliff hangers at the end of each episode so make sure you can control yourself before watching or you will be up for 24 hours watching it all.
K-On!: Cute girls doing Cute things. Loosely based on music.
Mirai Nikki: Action and stuff. It is good.
NHK ni Youkoso!: A.K.A Welcome to the NHK It's about a shut in and how he copes with his life. Deals with some deep issues facing certain parts of society
Nichijou: More humor from highschool girls. Different from Azumanga though
Panty & Stocking with Garterbelt: Looks like Cartoon Network, but it is very adult and amusing
Redline: Even if you just watch it for the art. It was 7 years all hand drawn.
Serial Experiments Lain: I don't even know what this is about apart from it has a lot of cyber references. It was very good though.
So Ra No Wo To: Another one that nothing really happens, but it has the best ending, in my opinion, which is weird, of any anime I have watched.
Steins;Gate: Time travel how it should be done.
Tengen Toppa Gurren Lagann: Robots that wont stop growing. One of the only Mecha shows I have enjoyed.
Toradora!: The greatest love story ever told.
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What our ancestors would really be thinking, if they were alive today, is: "Why is it so dark in here?" - Sir Terry Pratchett