Sword Art Online - the premise of this anime is that characters are trapped in a Full Immersion Virtual Reality video game, and have to defeat the boss to get out of it.  The creator of the video game announces that if they die in the game, they die in real life.  The anime follows main character Kirito on his quest to survive and win.
As for why I like it, the plot is great, the characters are well-developed, the world design is fantastic, and it makes me want to keep watching, to find out what happens next.
(BTW, beware of the end of the ALO arc - the bad guy then is a serious creeper.  Bleh.)

Detective Conan/Case Closed - the premise of this one is that the main character, Kudo Shinichi, is force-fed an experimental drug by a mysterious criminal syndicate.  The drug shrinks him down to a seven-year-old in body, but not in mind.  Shinichi has two main goals: don't let anyone know that he's Shinichi, and find the syndicate and take them down.  Unfortunately, there's the problem of murders literally happening wherever he goes.  Good thing he's a detective, huh?
As for why I like it, the characters are well-developed, the murders are plausible but not too gory, there's a good bit of humor to counterbalance all the death, and the writing is well done and realistic.
(Also worth noting is another anime that crosses over with DC sometimes - Magic Kaito.  It's about a magician thief who targets gems, seeking the gem that caused the murder of his father so he can destroy it.)

As for manga, I would recommend Kanata Kara, or, From Far Away.  The heroine, Noriko, is catapulted from our world to another due to a terrorist attack.  There, she meets Izark, the hero.  They go on an amazing journey, dodging bad guys who are searching for the Awakening, a being that is said to be able to summon the Sky Demon, which is the most powerful demon of all.
As for why I like it, the art style is good, the writing is great, the characters are varied and realistic, the plot is great, and Izark is really cool.~