What is Gauge Theory (intuitively)?

Many years ago, I was checking out at the cash register of a university bookstore, and a fellow graduate student in physics was with me in line. The cashier working the cash register saw the book I was buying was called "An Introduction to Gauge Field Theory" and made a frowny face, repeating the title to herself slowly and then asking "what's this about?!"

My colleague and I looked at each other for a moment, wondering how on earth we were going to sum it up before the other customers behind us in line got antsy. Before I could think of anything, my friend blurted out

"It's like, when you have two things that look different, but it turns out... they're actually the same."

She nodded her head, slowly, saying "okay", and we walked out laughing.

To this day, I can't think of a better way to sum it up. Although I would add that it's usually not just two different things, it's usually a whole parameter space of different theories that look different... but which actually describe the same physics.

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