What It Is
A piece of backpacking or camping equipment used to protect food and other edible-looking-or-smelling items (trash, toothpaste, sunblock, deodorant, etc.) from bears, and vice versa. Well-established campsites often provide bear boxes for this purpose, but further into the backcountry you're on your own. Bear cans are actually required in some areas where bears have learned to take down food hung from trees in bear bags. They're nearly unbreakable hard plastic containers operate on the principle that bears, though very strong and smart, do not have opposable thumbs. (If they ever do, we're in for a lot of trouble.)
How To Use a Bear Can
The lid of a bear can is easily locked or unlocked using a coin, screwdriver, knife blade, can opener, or any of a number of other simple tools. When you make camp for the night, stow your food and other smellables (again, this includes trash, sunblock, lip balm, bug spray, toothpaste, and any other scented stuff you may have with you) inside the can(s) and store these away from your campsite. That way, if a bear smells them during the night, it won't come searching through your tent (and maybe you) for food. My bear can came with a warning, which I'll pass on to you here: Never try to take a bear can away from a bear. I know that probably seems pretty obvious, but the fact that they had to put it on the package at all makes me feel okay about repeating it here. Finally, bear cans are pretty much guaranteed to survive at least one bear attack intact, but there's no telling if they'll make it through a second unscathed. So keep that in mind and try to keep your smelly messes to a minimum.
General Bear Safety Stuff
If you haven't already, go check out dmd's excellent writeup at camping in Bear Country. It'll give you a general idea of what we're working with here: smart wild animals who'd be better off not learning to forage for people food, because that habit will eventually get them killed. This goes for any wild animal, so for the love of whatever you care deeply for, don't ever feed one!
Although they very rarely hurt humans unless seriously provoked, black bears have been known to wreak major havoc on cars and other human property in their attempts to get at coolers, food, and other edible-looking or smelling items (their sense of smell is far better than ours, so don't assume that candy bar wrapper you think you licked clean isn't bear bait). The general human response to this and other aggressive animal behavior is to shoot the bear. So it's in your and the bears' best interest to protect food, trash, and other remotely edible looking and especially smelling items. Now, grizzly bears, they're even more dangerous than black bears, and they can't generally be scared away with loud noises. So everything I've said so far goes double for them.