What is the proper way to store garlic? Mines is in the refrig in a jar of oil.
In a cool, dark place. If you have a basement or pantry, that's best. And keep it in a basket or well-ventilated container.
Is this question about peeled cloves? If you bought them that way, store them as they were sold. Otherwise, separate and peel as needed from your 'cool, dark place.'
I store my garlic in a zip top bag in the freezer. It only takes a moment to thaw and the skin pops right off. I picked up the idea off a website from a garlic grower...
Responding to the second part of your query: I've heard that garlic should not be stored in oil in the refrigerator, due to the risk of botulism. There was a stern, rather frightening directive from the FDA sometime around 1990, after several people died from botulism from garlic in oil. I peel as I go, storing fresh garlic in the same small, mostly open storage crate that I use for my onions, which I keep on the floor of my (thimble-sized) pantry. ;o)