Brown and black bears are scattered throughout the United States. Grizzly bears are found pretty much from Canada to Wyoming. If you're going to camp in grizzly bear territory, I think you should really do your research, do your homework, make sure you're an experienced camper and make sure you know what you need to know camping in grizzly territory.
With brown and black bears, there's a lot of fear about them, but you probably won't cross paths with them, and bear attacks are really rare. The thing to remember about bears is that you should keep your camp site meticulously clean. You absolutely can't leave food around, trash around. Anything that smells or tastes like food, even toothpaste. All that stuff needs to be double bagged, tightly sealed and hung if possible. You should really hang your food.
The other thing to keep in mind is that bears that attack usually attack because people got too close, got too curious, taunted them. For the most part when it comes to wild life, if you steer clear of wild life and respect its territory, you probably won't have a problem.
If by some chance you do run into a bear and it seems like there's a potential for an attack, there's a couple of things you should do. Don't make eye contact with a bear. If you're near a bear, avoid eye contact and slowly start to walk, backing away. Keep yourself face to face with it and just start backing away, while still avoiding eye contact. Don't turn and run. Running is sort of an invitation for them to run after you. You can keep pepper spray around at your camp site if you really know how to use it. I think it might be a good idea. I've never camped with pepper spray, but it can be a deterrent if you're being attacked by a bear.
Bears will also often run away if you're loud and if you clank some pots. If you're at your camp site and you see a bear nearby, grab some pots and pans and just start making a lot of noise. They'll actually, usually, just start running away from you.
I think the biggest advice I can offer when it comes to bears is that you should just really do your homework. Make sure you know what the advisories are. Make sure you know what the rules are for where you're camping, and be as safe as possible. Make sure that you keep your camp site clean, you hang your food, and you don't go looking for trouble.