I would say this:
Design is the potential to communicate an abstract idea in a concise, and engaging manner. The potential the distinguish one idea from another. The potential to reach out to a specific, targeted group of people and evoke emotion, provide understanding, and shape an experience.
Graphic design is the set of rules and guides we employ to design, hopefully well. The arrangement of visual content in order to communicate a specific message using color, weight, balance, lines, shapes etc.
Past this, I often think of the difference between design and advertising as this: Advertising convinces you, design makes you understand.
A lot of gray area, though!
I also would say that Design is everything you love, and everything you hate. It's just as much about experience as it is about visual communication.