Before asking for anime recommendations for the 9000th time, make sure you've experienced the classics.
This site maintains lists of the current and previous season, as well as every series you ought to be watching, from the certified masterpieces and award winners to the popular and renowned.
Edit: Just for clarity,
you do not have to have watched all of these before you make a thread asking for suggestions. This is just designed to cover most of the suggestions that people would post anyway in a generic thread. If you have more specific queries, that's fine, but just make sure you've looked into the shows in that genre, by that studio etcetera beforehand. This is merely a handy list of things you might like.
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Spoilered below are a couple of the images to get you started. There are a few others at that link, but they're very similar and I know how lazy people can be. If you can't see the whole image because resizing is broken again, right click on it and either save it, or click:
Firefox - 'view image'
Chrome - 'open image in new tab'
IE - I think you have to save it to view it, because IE is retarded
General recommendations
By genre
Also, if you haven't signed up to
MyAnimeList (MAL), you might want to now. It's partly useful to brag about how much anime you've watched and to share with others, but it's also a great tool for finding new anime yourself, and collates a lot of information like synopses, ratings, reviews and the best sub groups. Fakku's own MAL group for discussion and list checking (independently maintained by Tsurayu) can be found
here.
This may have been done before, but you wouldn't have guessed.