Princess Mononoke is one of Miyazaki's most well known works, and is definitely worthy of viewing by even the most stout hearted of anime opponents.

The main character Ashitaka sets out on a quest to discover both the cause and the cure for a curse brought upon him after he slays what turns out to be a demon god.  As he travels he begins to learn that the curse is in fact related to man's continued use of technology to tame and conquer nature.  Ashitaka is then confronted with the choice of which side to fight for as the battle for the future existence of man and nature quickly takes shape around him.

 

At once a stunning and poignant tale of the struggle between man and nature, Princess Mononoke is a wonderful addition to any anime fan's collection.  If you haven't seen much anime, then this is definitely a good starting point.  It will show you that not all anime is for kids, especially not this one.  Due to some graphic imagery and violence, I highly recommend parents give this a viewing before letting their kids watch.  I expect it may even suprise you at how 'adult' anime can be.