Grizzly bears and brown bears are really the same bear, Ursus arctos, and they are the most widespread bear in the world today. They are found throughout the northern hemisphere in their preferred habitat: mountain forests, open meadows, and river valleys (though modern bears are also fond of garbage dumps). Once found as far south as Mexico in North America and Spain in Europe, today brown bears are largely confined to isolated pockets in remote areas.
Brown bears are heavy and stout with strong muscular legs, a big head, plantigrade feet (their heel and toe both make contact with the ground when walking), and a short tai]. They can run quite quickly over a short distance. They have a distinctive shoulder hump and large curved claws on the front paws adapted to digging for food. They have a thick shaggy coat which can be black, red, brown, blond, or a mixture of these colours. Adult bears are generally about 200 cm (about 7 feet) long and weigh anywhere from 150 to 375 kg (330 to 825 lbs), depending on their food sources. Males weigh twice as much as females, and males with access to rich coastal resources - those salmon-catching bears you've seen on National Geographic - often weigh 500 kg (1100 lbs). These giants are the ones often referred to as grizzlies. Brown bears are omnivorous, but 75% of the diet is vegetarian, consisting of seasonally foraged foods like berries, grasses, herbs, tubers, roots, and nuts. They also eat decaying animal carcasses, fish, marine mammals, insects, honey (of course), and a variety of small mammals like squirrels, marmots, and elk calves. Such large animals require a high caloric intake, and during summer and early fall will eat up to 40 kg (85 lbs) of food a day, gaining a few kilograms of fat each day. They are preparing, of course, for hibernation.
Hibernation is an adaptive process that allows these huge animals to survive cold winters when there is not enough food to sustain them. They hole up in a cave, hollow log, or tree cavity, or dig a den in earth or brush. There they will spend the cold months, with their metabolism slowed by half, their heart rate slowed to about 10 beats a minute, and their body temperature reduced by a few degrees. They do not pass any waste during hibernation, having developed a unique process of recycling their urine into usable proteins. Females give birth during hibernation, and emerge in the spring weak and very thin, with vulnerable cubs in tow.
The intersection of bear and human is one fraught with danger on both sides; every year a few people are mauled to death by a frightened or angry bear, though we kill more of them than they do of us. So please don't feed the bears. For tips on what to do to avoid being chased by bears when walking in the woods, see mattbw's useful bear attack. Or you could try a heavy and expensive bear protection suit, but I can't vouch for their effectiveness.
Information from "The Bear Den"