Here is a link for basic instructions. It's really quite easy and I've done it many times, once with PEACH peels & pits! http://www.vinegarman.com/VinegarMaking.shtml
You do NOT need any special equipment.
You do NOT need "a mother" as your vinegar will make its own.
I originally got my instructions for a book called Old Fashioned Recipe Book which I just love to pieces. For the vegans, it does talk about animal products, so be forwarned.
Vinegar is a two step fermentation. Tirst, one kind of critter turns the sugar into alcholol. This is why you 'could' use that bottle of wine you've not wanted to drink to make your vinegar! Then another critter takes over and turns the alcohol into acetic acid.... vinegar!
This could turn into a long post.. I'll try to abbreviate when I can.
What to start with: Vinegar flavors and acidities vary according to what you start with. Cider vinegar means you start with apple juice or peelings and corings. White vinegar starts with grain. You can make vinegar from any homemade wine but not from storebought because they contain preservatives. Some imported wines can be used if they don't have preservatives. You can make vinegar from juices like pear, grape, cherry, or peelings and corings of those, or from whole cherries. You can add makings every day for a while (sugar, molasses, juicees) and if it stops working before the vinegar comes, you can add more sweetening. If it won't bubble or you want to hurry it up you can add 1 tablespoon of yeast, but I'd suggest you first give it a chance.
WORKING IT: Put your startings in a crock, wooden container, or widemouthed glass jar. Those gallon glass jars are perfect. Never ferment in metal or chipped enamel. Leave about 1/4 to 1/2 your crock empty for later additions and bubbling up. Cover with a clean cloth and tie that around teh neck of your container to keep bugs from dropping in. Wash and replace the cloth once a week. Set the crock in a warm place - 80 degrees is a good temp. The cooler the temp the longer you will wait. By a stove or heater vent is good.
During the first stage, it bubbles. As long as that's going on, bubbles will be coming up. When it's done bubling it's ready for the second stage.
THE MOTHER: Mother is a thick, clear, jelly like sort of amber substance made by the kind of beasties that make the vinegar and they live in it. If you've never made vinegar before, you may have to wait up to 6 months for yours to be ready. Often, I find the mother in the bottom of my Braggs ACV, and you CAN add that!
The crock surface will gradually start to show a think gray film. DO NOT DISTURB IT! It's important that it stays on top until it's work is done. Then it settles to the bottom, heavy and thick.
Don't lose your mother once you have her! She will make your subsequent vinegar makings MUCH quicker! Store some mother in glass jars so next season you'll have her. Just put a bit in a jar, cover with vinegar, and cap.
You can strain the vinegar until it's clean enough to suit you. It will gradually become more mellow. This homemade veingar can be used as a starter for your next batch.
TO STRENGTHEN VINEGAR: Let it freeze on top, then tak ethe ice off the top.
PURE CIDER VINEGAR: Let homemade cider stand exposed to the air for a month or two as instructed above, and it will turn to vinegar. Tart apples make the best vinegar.
Next post will be recipes for various types of vinegar. Please feel free to ask questions. If I know the answers, I'll respond.