If you generally don't watch anime, or if you do and have a friend you wished would watch with you, then this is the gateway anime to use. Especially if you like shows like "The Walking Dead" and "Game of Thrones."
Attack On Titan just finished its first season and took the anime community by storm. It is easily the best anime of 2013 and arguably one of the best ever. Nearly every episode had some events or reveal at the end that made me swear at my TV for forcing me to wait till next week. If I were just starting it now though I would have powered through the whole season in a couple days.
There's no way I can think of to tell you about the story without spoiling something. There are more plot twists than M. Knight Sholomon could dream about. Here's a general background though which should reveal too much.
Several hundred years ago, humans were nearly exterminated by Titans. Titans are typically several stories tall, seem to have no intelligence, devour human beings and, worst of all, seem to do it for the pleasure rather than as a food source. A small percentage of humanity survived by walling themselves in a city protected by extremely high walls, even taller than the biggest Titans. Flash forward to the present and the city has not seen a Titan in over 100 years. Teenage boy Eren and his foster sister Mikasa witness something horrific as the city walls are destroyed by a Colossal Titan that appears out of thin air. As the smaller Titans flood the city, the two kids watch in horror as their mother is eaten alive. Eren vows that he will murder every single Titan and take revenge for all of mankind.
The show can be watched in its entirety for free on either Hulu or Crunchyroll.
I cannot wait for the second season next year.