For information on the health benefits of bone broth, please see the linked articles at the Weston A. Price Foundation or Kelly the Kitchen Kop. Sally Fallon and others of the Weston A Price Foundation have already said anything I would say, and definitely better and more knowledgeably than I would. Kelly has great links and tips and her commenters often add useful and informative tips.
To make my nutrient rich bone broth, I fill up my large crock pot and roast the bones for a day on low. Then I add water and keep on low for several more days, continuing to add liquid as needed. Your broth will develop a scorched taste if the water to meat fat ratio is too low. (Yes, I learned this by experience). Sometimes I also add whey or vinegar with the initial water to leach even more minerals from the bones.
After several days, strain first through a colander, then through a few layers of cheesecloth, and finally through several layers of cheesecloth. At each straining, pour some boiling water through afterwards to get all the good stuff into your broth.
You may pick through the scraps you strain off and salvage the meat for use, but if you do, I suggest rinsing in boiling water afterwards. After cooking for several days, the bones are very crumbly and although it's probably immensely healthful I don't like that gritty texture, so I try to rinse most of it off. Depending upon the recipe, however, this grit is more or less noticeable.
I suppose one could remove the meat after the initial roasting and before the "stewing for days" part. I guess this is kind of obvious, but I've never done it. I wonder how that would effect the nutritional value of the broth. I'm guessing here, so don't quote me, but I think the nutritional value of the broth is from the gelatin and the minerals in the bone. The meat is your protein, which is just as good roasted for one day as it would be for several. Anyone have any knowledge on this?
Once you have your broth clarified, store it in your fridge. You can use it for anything that calls for broth or cooking liquid for a nutritional boost that avoids the additives in pre-packaged broths. You will have to add salt to taste, either after the broth is done cooking, or to each recipe. I usually wait and add it as I use the broth, since I don't always want equal saltiness in different recipes. Also, because the broth is quite concentrated, I usually dilute it with equal parts water, or to taste. Again, it depends upon the use to which I am putting it.
I don't have any hard facts on this, but it keeps quite awhile. As long as the fat layer is intact, it seems to keep indefinitely. But even after that layer is disrupted, I've used it for several weeks or even a month if I don't use it up. Use your own judgment. I've also stored it in the freezer in small portions anticipating cold and flu season.
I store my broth in a gallon jar, and I try to take a proportional amount of the fat with each use, so the fat lasts until the final drop of broth.
For a turkey or chicken carcass, I do the same thing. Take it right off the table and put it in my crock pot. Obviously, it will not need roasting first; just add liquid and simmer for several days.
To be really thrifty, some people keep an ice cream bucket in the freezer for any bones from the nightly cooking and table scraps. When they have a bucket full, they cook it all into a broth. I am not that thrifty. If the need arises, I know those scraps, too, can be put to use.
I plan to save some crumbly bones to put on my garden this spring, too. There must be a few remaining minerals in them. I think they would work in a compost system also, in place of bone meal, but be sure to rinse them if you don't want to draw undesired critters.