Essentially, you take two stainless steel bowls of roughly equal size. Take a head of garlic. Smash it with the heel of your palm to break it apart into individual cloves. Dump the lot into a bowl. Cover the top with another bowl. Shake shake shake. I tried this at home with a couple of different things. Since the concept seemed to be tumbling garlic together, I tried this with a tupperware box. No dice. It seems like the round shape of the bowl keeps things moving. I tried this with a plastic salad bowl with lid. Still no dice. It needs to be /two/ bowls. I tried this with two plastic bowls. It was getting better, but I had a couple of cloves still tightly holding onto the skin.
Two stainless steel bowls, of comically large size later, and I had a head of perfectly peeled garlic. I think part of the peeling process is the smashing with the heel of your palm. I tried with just loose garlic cloves that were lying around, and they didn't work as well. The other part is agitating against the stainless steel bowl, which has a bit more grip than plastic or ceramic. It clings to the garlic, and not the skin. Also, with the cloves of garlic smashing around, and hitting each other, you've got a bit more abrasion going on than you would if it were one or two cloves of garlic.
Try it out, with two large stainless steel bowls, and you'll be happy that you did. Next step is to chop them. Or, to leave them whole. Or, to run them through a garlic press. I got one of these garlic presses a few months back, for a review copy, and have been impressed thus far. The cost seems to be fairly reasonable. The thing about it is that the OXO doesn't like unpeeled garlic. With this method, you can use the OXO, which does a fine job of getting the garlic into your cooking pot (rather than all over your counter, or stove, as the old style ones used to), and is easy to press without a lot of effort.
Frankly, I'm quite happy to use an entire head of garlic in my meals, but if you're not, there's a trick to storing peeled garlic. Just place an absorbent paper towel (lightly crumpled) in the bottom of a tupperware container that's just big enough to store the garlic cloves. Dump in the peeled garlic. Cover the lid tightly, and place it in your vegetable drawer of the fridge. It'll last a good few days, and won't get all wet and gross on you. Mind you, it is still best to use your peeled garlic immediately, but if you can't, you can buy yourself a couple of days of insurance.
If you do want to use the whole head, and don't want a SUPER strong garlic odour, just add it into your hot fat, at the beginning of cooking, rather than towards the end. Add it even before you add onions. The sharp bite will cook out, and the garlic will get more mellow. When you eat the final dish, it'll be garlicky without being overpoweringly garlicky.
Final note: do not let the garlic get more than a medium brown while you cook it in the hot fat. Burned garlic is bitter, and extremely unpleasant.