Oh man, where do I even begin...
I'm going to put down Bleach, Fairy Tail, Naruto, and One Piece as those anime that any anime/manga fan knows of. So I'll mention others. I didn't pay too much attention to what everyone has posted (overeagerness got the better of me
so I may repeat some already mentioned)
Death Note - A guy gets his hands on a notebook that whoever's name is written in it, they die. So he decides to kill all the evil people. Very mind-gamey, it's majoritly about a legendary detective trying to catch this evil guy and the mind games they play on each other as they try to outwit them.
Full Metal Alchemist: Brotherhood - Magic-esque alchemy, with alchemy on human beings a forbidden taboo. The characters are great, especially Armstrong. The whole system intrigues me, an equivalent exchange of one set of stuff to create another set. Militaristic political system too.
History's Strongest Disciple Kenichi - This anime stopped after 50 or so episodes, but the manga carried on much further. It's a martial arts story, but the masters break the bounds of human capability. Very very funny, one master is always drinking, another is a pervert, another is very philosophical with a sadistic streak in his training.
Hunter x Hunter - This.. is an interesting one. On the surface it's about a kid wanting to be the greatest Hunter in the world, Hunter being a member of an organisation of sorta bounty hunters, but they can also be body guards, archeologists and the like. But the show places a lot of emphasis on riddles and powers from a chakra-like source called Nen. I find the show makes me think a lot about the various tricks and puzzles used.
Sword Art Online - A bunch of people jack into an MMO of the same name using a virtual reality kit, it synchs up to your nervous system, so it's like you're actually there. Unfortunately they all get trapped there, unable to logout and are told if they die in the virtual world their VR kit will microwave their brains and kill them, only by clearing all the floor of the dungeon will they escape.