My brother Jason hit upon a great analogy for explaining cloud computing to our (nontechnical) parents late last night: it's like a laundromat.
You want clean laundry. You can put a washer and dryer in your own apartment and have access to it whenever you want, but you have to pay for the machines, maintain them, and are limited to one washer and one dryer load at all times unless you buy more machines. But why would you? That single machine is probably already sitting empty most of the time, because most people do a couple loads once a week.
But here's the thing - you wanted clean laundry, not a washer and dryer. And all your neighbors probably want clean laundry as well. And furthermore, you're likely to do laundry at different times; if you do your laundry on Saturday and Mr. Jones next door does his on Sundays, you and Mr. Jones could theoretically share the same washer and dryer and save money.
Scale that idea up into a business and you get the laundromat. It's got a lot of machines, so whenever you need laundry done, you can walk over, stuff them into unused machines, and pay by the load. You're no longer limited to one washer and one dryer; if your little cousin gets the flu and projectile-vomits all over the house, you can do 10 loads simultaneously, and you don't need to ask for permission to do it - you just pay for more loads. If you go visit your grandparents for a month, your laundry costs you $0. Not only do you not need to maintain the washers and dryers, you're likely to be using bigger, better, and newer machines than you'd get for yourself.
The laundromat is like a public cloud.
Okay, said our parents. But then why would anyone want a private cloud?
Think about printers in your office, I said. Having your own washer-dryer is like giving everyone in the company their own desktop printer; using the copy shop across the street is like the laundromat. But as the big boss, you might not want all your employees using the copy shop across the street because... gee, that's a lot of confidential information floating in a public place (whether your nervousness is justified or not). So what do you do? You get a giant super-printer, stick it in the office, and hook everybody up to that. That big printer is yours; it's in your office, on your network, maintained by your IT staff. The printer is like a private cloud.
And in practice, you're going to get a few employees with desktop printers, a bunch of big office printers that get heavy usage, and people running across the street to the copy shop when they're on business trips or in a hurry to get 10,000 binders made right before the giant Sales meeting. That's a hybrid model.
They got it.