Sourdough starter is wild yeast, cultivated in a mix flour and water. If you've ever dissolved some bottled, store-bought yeast in water and watched it puff up, then you know roughly what sourdough starter is like. The difference here is that you're feeding the yeast every so often with more flour and water to keep it going.
So: how do you go about acquiring your own starter?
Of course, you could order some starter online, or get some from a friend. But why let someone else have all the fun? The basic idea is to mix together roughly equal parts water and flour and wait for the yeast to take up residence. In practice, of course, there are a couple more steps you can take to increase the likelihood of success. Each baker has a different idea of what works best for capturing wild yeast, but here's the strategy I've settled on, combining some King Arthur Flour tips and experimentation:
Start out with three or four ounces each of whole-wheat flour and lukewarm water (either bottled water or the filtered stuff—yeast is a living thing, and just as you don't use tap water with goldfish, try to avoid it with yeast). Use a very clean glass or ceramic container, preferably one just out of the dishwasher or scalded with boiling water. A pinch of sugar or a drop of molasses or honey stirred in to the slurry will give any yeast stumbling upon the slurry an extra jolt of energy. Cover your container loosely and set it in a warmish spot.
The next day, scoop out about half of the slurry and throw it out. Replace what you just threw out with equal parts whole wheat flour and purified water, stirring to mix it into what was already there. Leave it, again, covered in a warm spot. Repeat this for a few days. You'll start to see some bubbles in the slurry if it's working, and you'll be able to smell the yeast. At this point, switch to feeding twice a day with white flour and water instead of whole wheat. After about a week, you should have a very well-established colony of happy, healthy yeast. If not, try again! It's a low-energy experiment. Of course, there are always other methods, as well as the ordering route. To build up the amount of starter you have for a recipe, just add more flour and water during feeding than you discard.
Now that you have your starter:
At this point most home bakers go low-maintenance, feeding the starter only once a week. To do this, feed the starter as usual by discarding at least half (or as much as all but about half a cup) and replacing the discarded starter with flour and water. Leave it out on the counter for about an hour, and then stick it in the fridge for a week to slow the yeast down. When a week has past, you feed it again. You may notice the starter settling down into a dense layer, with alcohol pooling on top. That's completely natural, as alcohol is a byproduct of fermentation. Stir it in, discard part, feed. A little bit of grayness is also normal in tired, hungry starter after a week in the fridge.
Only start to worry if something smells foul: if things start to go wrong—most likely because you've managed to kill off your yeast, whether through starvation, contamination, or temperature extremes—you'll smell it, and likely see it, too. Healthy sourdough has a sharp scent, sort of like that store-bought yeast as it's bubbling—except with more alcohol smell, and a hint of wilderness. Though you may get some mold up the sides of your sourdough container, where accumulated moist flour is but yeast isn't, all you have to do is dump out your starter into a bowl, clean the container, and put the starter back in.
Many recipes start out demanding "ripe" starter, or starter that's been fed and allowed to go wild on the fresh slurry until it reaches its peak level of bubbly expansion. This is the only part about starter that's hard—volume varies with ripeness, and so it's better to measure by weight. As I said, though, starter is very forgiving; if you're trying to bake bread, it helps to use ripe starter, carefully measured. You can, however, make fantastic pancakes and quick breads with the starter you take out of the container during feeding, just using some imprecise, old-fashioned measuring cups.