Vinegar happens naturally, given the right conditions; this is the reason that most apple juice is pasteurized, and sulfites are put into wine - to stop the process.
To make ACV I bought some apples and ran them through my juicer (I would have bought unpasteurized cider, had it been Autumn). You wouldn't want to use a steam juicer for this, as you don't want to heat the juice. Lacking a juicer one could run them through a food processor or blender. Depending on how they are juiced, and how much pulp is still present, one might want to strain it - dump the lot into a jelly bag or clean pillow case, suspend above a container, and leave it for a day. The point is to get about half a gallon of reasonably clear juice that has not been heated. If the apples were not sprayed I would just give them a quick rinse, and then proceed. If sprayed I would scrub a bit harder, or maybe even peel them.
Then I put the juice in a gallon jar (I filled it first with boiling water to get rid of any mold spores etc.; one could also boil it for awhile, as for canning jars), with a wide mouth, only filled about half way. A large surface area exposed to air is necessary (OK, maybe not
necessary, but desirable). Then I covered the jar with a clean cloth, and left it on the kitchen counter (and put a pretty doily over it, so it wasn't quite so unattractive

). Every day or so, or when I thought about it, I would swirl the contents about a little, to introduce a bit of oxygen, but otherwise I just left it to do it's thing. The juice will naturally form alcohol, which will naturally turn to vinegar. The process will take several months. Its done when it smells like vinegar. It does NOT need to be heated to stop the process; vinegar is an end product, it won't turn into anything else. It may or may not form a mother - an icky looking disc floating in the vinegar. It doesn't matter; it will turn to vinegar even without the mother, and will still have all the good bacteria. I was making vinegar for several years before I ever saw a mother.
The reason I started with ACV is because it is foolproof. Apple juice
will turn into vinegar unless specific measures are taken to prevent it. Once one has the vinegar going it can be used to start other batches, even without a mother. Make or acquire some more juice (this time one could use pasteurized juice, as the proper bacteria will be introduced from the previously finished vinegar), and mix a cup of the finished ACV into it (or the mother, if one formed). Treat it just like the first lot, but the process should go faster. (The initial project will also go faster if one has a cup of unpasteurized vinegar to jumpstart it - Braggs or similar - but it isn't necessary)
Once I had the finished ACV I started some wine vinegar. As stated, wine usually has sulfites to stop the vinegar process from starting; wine will usually just go bad if left in an open container. But I started with some inexpensive red wine, put it in wide mouth jar, and added a glug of the new ACV - and ended up with red wine vinegar with a touch of apple. I've used this to start other batches, and the apple is no longer apparent. White wine generally has more sulfites than red, so I did the red first, then added a glug of that to some white wine, and ended up with white wine vinegar tinged with pink (and a very tiny amount of apple

). I have also made a malt vinegar with a bottle of local micro-brew - this was when I finally got a mother culture; seems that malt vinegar is more likely to form one.
I have, on a very few occasions, got mold forming on the top of the juice. When that happens I toss it out, though it has only happened a couple of times. Both times I tried to make the vinegar in a jug with a narrow neck, so I believe that there just wasn't enough air exchange going on. Use a wide moth jar, and clean/sterilize it, and mold should not be a problem. And once the vinegar is ready transfer it to a regular bottle with a tight cap. It will keep "forever", and be there ready to start a new batch as needed.
Disclaimer: Canning and pickling recipes will specify that one should not use homemade vinegar when canning/pickling because one can't be sure of the strength. So take that into consideration when using your homemade vinegar. Personally, I don't worry about it that much; I figure my vinegar is as it was made, and not "diluted to 5%" as commercial vinegar is; and many pickle recipes say to dilute it even further. So I have used my vinegar on occasion - but that is only MY way of doing it, and I don't recommend it.
P.S. -- For the
really frugal among us I will add that some people have made ACV just using the peels and cores of apples - after an applesauce making session, for example. Google should help you find recipes for that if you like. I haven't done it, so can't say how strong it would end up being, as one adds water to the peels, and goes from there.