Yeast starters are a liquid with near-ideal conditions for yeast growth, used to grow a small amount of beer yeast into a larger quantity suitable for pitching into a batch of beer. Both under- and over-pitching yeast into a beer can produce off-flavors, so creating the right sized starter is important. Packaged yeast also tends to go dormant, and a starter revives them.
Starters are typically made with dry malt extract (DME) to ensure that the yeast that thrive in the starter are used to a wort-like environment. The target specific gravity of a starter is typically 1.040. This is obtained with a 10:1 ratio of water to DME (i.e. 100g DME per liter).
To avoid 'stressing' the yeast, starters usually start small, and are increased in size every day or two, by a factor between 2x and 10x the volume. One recommended pattern is vial/smackpack → 500ml starter → up to 4L starter. The starter can optionally be cold-crashed (chilled), the wort decanted off the top of the yeast cake, and fresh wort added onto the cake. Doing this method requires less total volume starter liquid per hundred million yeast.
The final starter can be pitched completely into the fermenter, or decanted so just the yeast cake goes in, depending on whether the volume of starter is likely to affect the flavor of the final beer.
Oxygenation is critically important to yeast starters. No airlock is used, and they are aerated frequently by shaking or by a stir plate. They still need to be protected from the environment, so a piece of plastic wrap over the top with a rubber band is an option.
A “vitality starter” is created by pitching a pack of yeast into 500-1000ml of 1.040 SG wort and aerating it extensively for 4 hours (on a stir-plate, theoretically), then pitching it into a non-oxygenated fermenter. The theory is that the yeast count is lower, but the yeast are in their peak oxygen-consumption phase, before they start creating alcohol, and will rapidly get started in the full wort. This seems to be widely adopted by homebrewers now, and the main advantage is that you don't have to prep a starter days in advance – just prep it at the beginning of your brewday.