In a small bowl or glass liquid measure, put 1/4 to 1/3 cup plain yogurt. Set aside with a whisk and rubber scraper.
Prepare incubation method (I have a yogurt "maker", so I plug it in to warm and get out 6 yogurt cups).
Heat 1 quart of milk in a saucepan, covered or not, using a candy thermometer to check that it reaches 180 F and checking/stirring often.
Sit the saucepan in a cold water bath to cool to 105 F, still using the candy thermometer to check.
Pour approximately one cup of the cooled milk into the starter yogurt. Whisk well. Pour the combined starter and milk back into the saucepan and mix everything, again well. Use the rubber scraper to make sure you're not leaving behind the good stuff in the saucepan and bowl.
Pour the starter and milk into prepared containers, with loosely fastened lids, for incubation. Incubate at 110 to 120 F for at least 2 hours, usually 3 and up to 5. When it is jelled, refrigerate immediately to stop the bacterial growth (the longer yogurt bacteria grows, the more tart and firm the yogurt).
Notes on the Method:
There are tons of ways to incubate yogurt - explain your method in the comments if you wish, to help along people that don't have yogurt makers.
Most recipes tell you to sterilize everything you are using to make yogurt. I don't bother: all my things are just clean from being washed with all the regular dishes. If you are having trouble getting yogurt to set, try using boiling water to sterilize all the utensils and containers you are using.
Making yogurt with skim milk will give thin, watery yogurt which is fine for baking; for eating straight up, I'd recommend at least 2% milk.
When you heat the milk at the beginning, you are killing any potentially competitive bacteria, so don't skip this step or you may end up with just spoiled milk, not yogurt (yup, I learned this firsthand); you will introduce the desired bacteria in the starter.
I make a cold water bath by plugging my sink and putting in a few inches of cold water and a tray of ice cubes. You can allow the milk to cool without the cold water bath, but it takes a lot longer.
One time, irritated with a batch that wasn't setting, I went to bed and checked it in the morning. It had set! Usually, however, my yogurt sets in 2.5-3 hours.
How I use yogurt:
In recipes where sour cream or buttermilk is called for. To use yogurt for buttermilk, you may want to add a tablespoon or two of milk per cup to get a more accurate consistency.
To make salad dressings.
Smoothies - or popsicles.
Occasionally as a snack with fruit or granola.