Add the apple pieces and sugar to a large sterilised glass jar. Top with the water and vinegar, and stir well with a wooden spoon to dissolve the sugar.
Place a weight on the apples (like a clean stone or heavy small dish) then cover the jar with a folded tightly woven tea towel or several layers of cheesecloth. Secure tightly with a rubber band.
Set the jar in a cool, dark place with good airflow to ferment for about a month, stirring every few days or so. It should start to visibly ferment after a few days.
After a month has passed, strain the solids out through a fine mesh sieve and pour the liquid into a new clean jar. Cover tightly again and ferment for about another month. It should smell strongly of vinegar.
Transfer the vinegar to a clean glass bottle or another jar with a tight fitting lid and store like any other vinegar.
Notes
Sterilise the jars you use for the vinegar either with the oven method or running through a hot dishwaser before use.Be sure to cut out any bruises or buggy bits in the apples before use.The amount of liquid in the jar will naturally decrease over time as it evaporates, so you won't end up with a full two litres of apple cider vinegar.You'll almost definitely get something that looks like a kombucha scoby on your vinegar. This is good, and normal, and something we like to see! That's the mother, also known as a vinegar scoby.Anything can be used as a weight, just like you would for something like sauerkraut. A non-porous stone that's been boiled or run through the dishwasher, some kind of small dish that can fit in the jar, or a purchased fermenting weight. Apples that are exposed to air will form mould.Sometimes fruit flies can get in to the vinegar and that's perfectly fine. You can fish them out, make sure the jar is covered more tightly, and let it continue fermenting. If you hate the idea of it then make sure you use several layers of a tightly woven cloth and check that it doesn't have any small holes in it. They've only gotten into one batch I've ever made.