Sourdough is a term used mainly in the U.S. to describe bread which is leavened naturally (i.e. without the addition of commercial baker's yeast) before baking. The best know example is the famous San Fransisco sourdough, which is, as its name suggests, unique to San Francisco, California. A sourdough starter is simply a culture of the various microorganisms that are used to leaven sourdough bread and give it its characteristic flavor.
Prior to the 1860s, commercial baker's yeast was unknown, and all breads were leavened naturally, whether by the methods described here or by the addition of another source of yeast, such as ale barm. (Note that prior to the 20th century, all beer was also fermented naturally, and much of it was also sour.) Commercial baker's yeast is propagated from various strains of the species Saccharomyces cerevisiae, selected for high leavening power and neutral flavor. In contrast, natural leavens (i.e. sourdough starters) contain a variety of "wild" yeasts which produce carbon dioxide to leaven the bread, and acid-producing bacteria which (typically Lactobacillus and Pediococcus species) that lend a distinctive flavor to the bread and also contribute to leavening. In San Francisco sourdough, for example, the dominant yeast is Candida milleri and the main bacteria species is Lactobacillus sanfranciscensis, also referred to in some sources as L. sanfrancisco. German and Russian rye bread starters often contain species such as C. milleri, C. krusei, and S. cerevisiae, plus a veritable zoo of Lactobacillus and Pediococcus species.
From the description above, it should be clear that a proper sourdough starter cannot be made with commercial baker's yeast, and any method suggesting otherwise is to be regarded with suspicion. Likewise, methods that suggest leaving a mixture of bread flour and water to collect wild yeast are unreliable at best. There are however, three extremely reliable methods available to the average home baker. The first of these is to purchase a commercial sourdough culture. These cultures are available by mail-order from several companies, and are typically priced between $10.00 and $20.00 U.S. While this may seem expensive, keep in mind that it should be a one-time investment as your starter can be propagated ad infinitum if you follow sound methods. This method is especially attractive if you are trying to reproduce a certian style, such as San Francisco sourdough, because these commercial starters are pure cultures of the appropriate organisms. The second method is to acquire a sample of starter or dough from a local bakery that bakes sourdough bread. Many shops will be willing to give you a sample free of charge. The third, which I personally prefer, will be described below, and requires only two ingredients: water and whole-grain rye flour.
The water and rye flour method (whole wheat flour will work too, but whole-grain rye is preferred), takes advantage to the fact that whole-grain rye is "contaminated" with various Candida and Lactobacillus species, and also contains abundant nutrients that allow these organisms to flourish. To make a starter using this method, simply combine 1/4 cup of whole-grain rye flour with 1/4 cup of chlorine-free water in a glass or plastic container and mix thoroughly. Cover loosely and place the container in a warm (~80°F) location. Within 24 hours, fermentation should be evident (the mixture will be foamy). At this point, the volume of starter is increased by adding one cup of rye flour and one cup of chlorine-free water, and the mixture is again allowed to sit until fermentation becomes obvious. The starter should have a pleasant, acidic odor and flavor at this point. If it smells moldy or musty, or if mold is visible, discard it and start over. This mixture may then be stored in the refrigerator until needed. The character of your starter may change over the first several weeks, but it will eventually stabilize.
Once the starter is established with rye flour, you can use it to innoculate other types of flour as starters for different styles of bread. For instance if you want to make white bread, prepare a mixture of equal amounts of white wheat flour and non-chlorinated water, and add a tablespoon or so of the original rye starter. In this way you can make starters for any style of bread you like.
Because the starter contains living organisms, it must be refreshed periodically or its nutrients will be depleted and the culture will die. If you bake often this is simply done by replacing the starter you use with an equal amount of flour/water mixture. If stored in the refrigerator and not used for an extended time, you should periodically (every two weeks or so) discard half of the mixture and replace with an equal amount of flour/water. After doing this, the starter is allowed to sit for 24 hours in a warm place before being returned to the refrigerator. This process is often referred to as "feeding" the starter. Your starter can be kept alive indefinitely using this method. If you bake often, another method is to simply save a small amount of dough from the previous batch, and use it to start the next batch. This method is popular in french bakeries, where this piece of starter dough is called the chef. It is really only useful if you bake on a daily basis, however. For long-term, maintenance-free storage, a small amount of your starter can be mixed with ~10% glycerine and frozen for up to one year.
Metal containers should not be used to store starters, as the acidity of the starter will eventually damage the metal, and the products of the reaction between the metal and acid can be harmful to the organisms in the starter, and possibly to humans as well. Also, it is important to always use chlorine-free water when making and feeding the starter, as chlorine from tap water will eventually build up to levels that inhibit the growth of the starter yeasts and bacteria.
To make bread, typically 20-40% of the flour for a given recipe is mixed with an equal amount of water to form a loose batter. To this batter, a small amount of sourdough starter (1/4 cup or so) is added, and the mixture is left to ferment for a period of time (typically 8-24 hours). This mixture is called the sponge by American bakers. Greater percentages of flour and longer fermentation times lead to greater sourness in the finished bread. After the mixture has fermented for the desired period of time, the remainder of the recipe's flour and water and other ingredients is added, and the dough is kneaded in the usual manner. The actual formulation of the bread recipe is beyond the scope of this writeup, and is left as an exercise for the reader.