Action: Fate/Zero, Darker Than Black, The Legend of Legendary Heroes, K project, Hamatora
Romance: Clannad, Mirai Nikki (Also action), Sword Art Online (Also action), Toradora! and also check out any/all movies made by Makoto Shinkai
Comedy: No Game No Life, Great Teacher Onizuka
Horror: Another
Other: Death Note, Code Geass
If I had to pick out good animes to watch out of these, watch these.
Clannad + Afterstory - One of the best animes ever, without a doubt.
Fate/Zero - Amazing plot, action scenes and animation.
Darker Than Black - Amazing action scenes and plot (IMO).
Sword Art Online - Not for everyone, some find it amazing some find it bad, I find it amazing.
Mirai Nikki - Also not for everyone, it involves a Yandere girl and a sissy main character.
No Game No Life - For laughs, I'm not sure if you can find any other anime that's funny like this one.
Death Note - I think it speaks for itself if you hear this name.
Code Geass - Haven't watched it but it's also really good apparently.
Edit: Steins;Gate and FMA + Brotherhood are also among one of the best animes ever, I haven't seen any of them yet but I think FMA is action, I don't know anything about Steins;Gate.