My view is, as a BSEE educated in the 1970's about both analog and digital circuits, who saw analog computers in use in academia and in the factory, who worked on one or two of them in my working lifetime. The hardest thing about explaining analog computing, is that it's almost TOTALLY EXTINCT today. It's hard to think or talk about something, outside your experiences. And preservation of extinct computing - and explaining it in it's own context - matters to me.
One can explain "voltages not bits", in any number of ways, appropriate to audiences and settings. "Temperature" might be a good example. There's still mercury thermometers, mechanical thermostats and mechanical thermometers. They use the expansion of metals in an ANALOG way, to represent "temperature". The Weather Channel, gives it some number. But we feel "hot" or "cold" without numbers in our heads. A point to consider: the physical world is not lists of numbers, it's events and materials and the flow of time.
"Differential equations" is jargon used by engineers and scientists. But it comes down to actions in systems over time. How fast does water boil on the stove? Or freeze if put outside in the winter? How long does it take your car to go from zero to 60? How long does it take a baseball/football in flight to be caught? These are ANALOG values that change with time. And, there are analog circuits (components wired up) which can represent these rates-of-change calculations. Then you feed those circuits the analog values as voltages, and watch what they do over TIME.*That* is what an analog computer is about. Circuits and components, wired for one computation, using time and voltages to represent values - no "bits", no digital (except for numeric results or inputs).
I myself, don't tell people "this is too complicated to explain". I give them*something*; it will provide some impression and place to start; further consideration will be their choice to make.
So. I hope the two explanations above, of "voltages not bits" and "differential equations", are helpful. I hope the notion of "extinct computer preservation" gets some attention. Otherwise, I'm not going to debate some ultimate description or how-to-talk-to methods. I've explained why, and I've offered my own views.
Excellent answer. Thanks Herb. I'll throw in another example: volume knobs. Many (most?) car stereos still have knobs you turn to adjust the volume. It may be easy to tell a kid that's analog -- a directly relationship between mechanical movement of the knob vs. increase/decrease of the volume. Whereas if you use a software control to adjust the volume, even if it the interface is a slider or knob, that's digital.