I've read through this thread and thought I'd add to it a bit ...
As most have noted, stocks, classically speaking are made primarily with bones, mirepoix (onions, carrots and celery in a 50/25/25% ratio by weight), and a standard bouquet garni or sachet d'epices (basically, a bundle of herbs, usually bay leaf, thyme, parsley stems, cloves, peppercorns, tied in cheescloth and placed in the simmering stock). The basic ratio for brown and white stocks is 8 pounds bones, 6 quarts water, and one standard sachet/bouquet garni, as well as 1 lb mirepoix (8 oz onions, 4 oz carrots, and 4 oz celery).
Again, as most have noted, broths contain more meat than bone, and thus do not convert the collagen into gelatin to the extent that a stock will. The rest of the ingredients are primarily the same. Broths aren't usually made with salt as part of the cooking process, but of course have salt added for seasoning when used in the creation of a dish.
The big difference between the two is in their application as well as texture. Properly made stocks are gelatinous (and this is definitely what one wants) and I would submit more unctuous in aroma and flavor (the four hallmarks of a great stock is color, clarity, aroma, and body) than a broth. Stocks can be used to make soups, stews, etc., but they are essential in the creation of demiglace, where the quality of the stock directly affects the finished product. Broths will have a much "meatier" flavor, and due to this are not used (in professional kitchens, anyway) in the same manner. For example, I would use a brown stock to make sauce chasseur, but not a broth as the broth's "meatiness" could overwhelm a stock. I would happily use broth, however, for things like stews, soups, etc.
All this being said, in professional kitchens stock is much preferred to broths by most chefs/cooks as it is more "neutral" and can be used in a lot of different ways. I do not remember once using a broth while at cooking school; we made and used stocks for everything. Many home cooks unfortunately don't have this option, and rely on store bought broths sold as "stock," which in the strictest sense they are not (the addition of salt is a telling indicator). However ....
For the home cook, stock making is relatively straightforward and easy. All one needs is a large pot (at least 12 quarts or more in capacity, and with a narrow top if possible), the ingredients, a standard skimming/degreasing set up, a ladle, a roasting pan (if making brown stock), a strainer (a china cap or chinois lined with moistened cheese cloth is best, but a decent strainer will do as well), and a container in which to strain the completed stock. I won't bore folks with recipes (there are thousands out there, and they're all more or less similar), but there are a few things, should one desire to make their own stock, to keep in mind ....
1. Rinsing the bones - some chefs do this, some don't. Those that do believe it removes any blood or other impurities on the bones. Others don't bother. Totally personal preference on the part of the cook.
2. If possible, cut the bones into 3" pieces. This aids in the extraction of collagen/gelatin as it increases the surface area of the bones.
3. Never boil your stock if you wish to obtain a gelatinous stock. Escoffier's recipe instructed the cook to boil the stock, but this coagulates the collagen prematurely, making it very difficult to extract it from the bones and turn it into gelatin. Simmer it, with the occasional bubbles rising to the surface. The French call this a "smiling" stock.
4. If making brown stock, there are different ways to add the tomato product (tomato puree, paste, etc.). One way is to "paint" the bones with tomato paste, known by the term pince in classical cooking. Many chefs do this part way during the process of roasting the bones, allowing the tomato product to caramelize without burning (normally, the bones are roasted in a fairly hot oven, 375-400 degrees). Other apply it earlier in the cooking process. Another method is to add the tomato directly to the roasting pan after the bones have been removed, pan degreased, and mirepoix browned on the stovetop in the roasting pan; this allows the tomato product to caramelize as well, lending flavor and color to the stock. Both are viable options and individual preference.
5. Some chefs add the mirepoix later in the cooking process, others add it at the beginning. I've done both with acceptable results. The same goes for the sachet/bouquet garni.
6. Keep the bones covered with water, adding warm/hot tap water if needed during the simmering process.
7. Once the the stock is simmering, move the pot off the heat about a quarter of the way. This is called mijoter in French, and it allows a convection to occur in the pot, making skimming/degreasing much easier.
8. Set up a degreasing (degraisser)/skimming (depouiller) station consisting of two containers one with water in it, and the other without. Skim the stock with a ladle, submerging the lip of the ladle just enough to skim off the solids and fat that will rise to the surface of the stock as it simmers. Do this occasionally, rinsing the ladle in the water to clean off the ladle before putting it back in the stock - this way the solids/fat recently removed will not be added back into the stock.
9. Professional kitchens simmer stocks for hours and hours, sometimes all night long. The general rule is for brown and white stocks, simmer for 6-8 hours (I normally go as long as 10-12), for fish stocks about 45 minutes, and vegetable stocks/broth abut 45 minutes. I've found that the longer I simmer, the more gelatin I extract. Totally up to the individual cook.
10. Strain using the cheesecloth/strainer/chinois/china cap. Don't press down on the bones/vegetables as it will cloud your stock. Ladle it out into the strainer. The stock can also be siphoned out with a rubber hose, or, if you have a spigot on the bottom of the pot (I had a spigot installed on the stock pot I use at home), merely drain it through the strainer set up. Shock the stock (refers to cooling in an ice bath) by placing a rack in the bottom of a stopped up sink, placing the top on top of it, filling the sink with ice and adding water to it until the ice comes up to just below the level of the stock in the pot. Using a ladle or spoon, stir the stock until it reaches the temperature of 70 degrees F. One doesn't have to stand over it until it cools - one can stir it as one passes by the kitchen while doing other tasks. Once cool, place in the refrigerator to cool completely.
11. The next day, remove any remaining fat which may still be present in the stock simply by lifting it off (it usually a solid disk that is easily removed). Use as needed, or store either by freezing or in the refrigerator.
Finally, I would not add a whole, scrambled egg if a clarified stock (known as a consommé, with additional flavor added via ground meat, vegetables, herbs, etc.) is desired. The yolk will irreparably cloud the stock (not to mention coagulate), and if one's desired result is a clear stock, this is probably not a good idea (although it would make a great base for a hot and sour soup). Use egg whites instead.
And I'm spent ...