A yeast starter is a solution that enables yeast cells to propagate (or reproduce). It resembles a mini batch of beer, but is not meant for drinking. Instead, you add it to your wort to ferment a better final beer.
Why make a starter?
The fewer yeast cells available for fermentation, the more likely you are to end up with stressed yeast. Stressed yeast, like stressed human beings, are the result of too few cells assigned to too much work. Your beer will ferment, but it will be of a lower quality.
Conversely, more cells equal better fermentation. Yeast starters increase attenuation, or the efficiency with which yeast ferments the sugar in your wort.
How does a starter work?
The secret to a yeast starter is aerobic metabolism, or keeping oxygen in the starter so that the yeast ferments very little, mostly just increases in cell count. For mail order customers, yeast starters also have the advantage of "proofing" the viability of yeast before brew day.
You will make a "mini wort" by boiling dried malt extract with water, cool it, put it in a flask, and pitch your yeast into it. A foam stopper will provide some protection from infection, but still allow oxygen into the starter for adequate aeration. Best results can be achieved by using a stir plate, which will keep the yeast in suspension and increase activity, and your yeast will be ready in less time.
If you are unable to use a stir plate, giving the flask a good swirl from time to time is advisable. It will also give you a good indication that the starter is going, since CO2 bubbles will come out of solution as you swirl it around, provided your yeast starter is in good health.
24-48 hours is usually sufficient time for yeast starters (although some strains like lager stains can take a bit longer to propagate). It is best to pitch the yeast at or just after peak activity, when your cell count has increased substantially, but while the yeast is still active. This will ensure a healthy fermentation in your beer, and decrease lag time before fermentation begins.