I think the challenge is not so much to explain how programming works (I think most have some vague idea that it involves making the computer do "stuff") but to

I think the challenge is not so much to explain how programming works (I think most have some vague idea that it involves making the computer do "stuff") but to explain why it matters to do it well and why you get wound up about seemingly incomprehensible stuff. I use the furniture making analogy; if you need a chair, a whole bunch of people can make you one. Some will make you a crappy chair from 2x2; some can make you a gorgeous looking hand-carved work of art; some might make you a modern sleek chair. While I might prefer the Louis XVI style chair to the Ikea style chair, the fact that the chair should not collapse when I sit on it is an absolute requirement. Hence, some people are great programmers and some people are bad programmers, and even the great programmers often disagree on questions of style.