Wash the apples thoroughly. Cut them into medium chunks, including the cores and peels. The more surface area you expose, the faster the fermentation.
Dissolve about 1 cup of sugar in 4 cups of water. This will help feed the natural yeast on the apples.
Place the apple chunks into the sanitized 1-gallon jar, filling it up ¾ full. Pour the sugar over the apple chunks, then pour the water over the apples and sugar, making sure they are completely submerged, leaving about a 2 inch headspace for stirring. You can add more water if needed, ensuring that the apples are fully covered.
Use a long wooden spoon to stir the apples. (I use the handle end to stir.)
Cover the mouth of the jar with cheesecloth or a clean dish towel and secure it with a rubber band or string. This is the best way to allow air to flow while keeping out dust and insects such as fruit flies.