I found this great article from AllRecipes.com, which is where this information came from.
Homemade stock provides a background to soup, so the ingredients you choose should be supportive, not overwhelming. Yesterday’s, or even last week’s, vegetables are fine, as long as they’re still healthy. The beauty of stock ingredients is the ideal ingredients are usually the trimmings from the soup you’re about to make (leek roots and leaves, tiny, end-of-the-head garlic cloves, potato parings, celery leaves, parsley stems, etc.)
Use a stockpot that is tall and narrow to help slow water loss from evaporation. To extract the most flavor from your stock ingredients, start with cold water. Meat stocks benefit from long, slow cooking. Vegetable stocks do not. Quick vegetable stocks should take 25 to 30 minutes; basic vegetable stocks, 45 minutes to one hour. Chicken or beef stocks can take anywhere from one hour to five–longer if you’re using a slow cooker. Certain herbs and vegetables will turn bitter as they steep. Strain as soon as the stock is finished.
Dos and Don’ts
Some vegetables should be avoided altogether in stocks. The cabbage family (turnips, rutabagas, Brussels sprouts, broccoli, cauliflower, etc.) does not do well in stock. Nor do most powdered herbs, ground black pepper, artichoke trimmings, or too many greens. Use whole peppercorns and bay leaves, as these are easy to strain out.
Cool your strained stock by filling the kitchen sink with ice water. Place the pot in the cold water bath, and stir every once in awhile, or until the broth is tepid. Store in the refrigerator for up to five days, or in the freezer for up to three months.