Pour the cider into a clean fermentation vessel, and allow it to come to room temperature if it was chilled. Be sure the cider doesn't contain preservatives or other ingredients. (I'm using a wide mouth carboy with an airlock, but any clean, non-reactive container will work.)

Dissolve a packet of wine yeast or cider yeast in a small amount of water. Allow it to rehydrate for 10 minutes before adding it to the cider. (Do not use bread yeast!)

If fermenting in a carboy (recommended), then cap the container with a water lock and allow it to ferment for 7-10 days at a cool room temperature (Roughly 18 C or 64 F). A water lock is optional, and you can just cover your fermentation vessel with a towel (but don't cap it as CO2 needs to escape during the fermentation process).

(Optional) To stabilize the hard cider, pasteurize it at 158 degrees for 15 minutes. This is optional but will ensure that the yeast doesn't consume all the sugar so that a bit of residual sweetness will remain for flavor. (Pasteurizing here only removes yeast, your finished cider will still contain the vinegar mother, which is added next.)

Add raw cider vinegar with the mother from a previous batch or from a store-bought container. There's no exact amount, but at least 1/2 cup per gallon, and as much as 4 cups per gallon of hard cider.

Secondary acetic acid fermentation needs a container that maximizes airflow, so choose something with a wide mouth. Use a non-reactive material, such as glass or ceramic, and avoid metal. (A mason jar, fermentation crock, wide-mouth gallon jar, or even a large ceramic bowl work well).

Secure a cloth over the top of the fermentation vessel to keep dust and fruit flies out.

Place the container in a room temperature (60-70 degrees) room, out of direct sunlight, and allow the mixture to culture for 3-4 months.

Once the vinegar has finished, it should have a cellulose mass at the top. This is the vinegar mother, and is completely normal. Strain and bottle the vinegar, reserving the mother to start another batch later.

Fully fermented apple cider vinegar should keep in a sealed container at room temperature for extended periods (years). I'd recommend flip-top Grolsch bottles for storage.