Here's how I make them, Pat, and they are really good.
Put your beets in a pot of water and boil until they are fork tender. Put on a pair of latex gloves if you don't want your hands to turn red, and rub the skin off under cold running water...it will slip off very easily. Slice the beets if they are large, and pack in sterilized jars. I really love the little baby ones.
Mix equal parts white vinegar, sugar and water in a kettle and bring to a boil. Sometimes I add a bit of mixed pickling spice to the vinegar mixture. Let simmer a few minutes until sugar is dissolved. Pour hot liquid over beets, put on lids and put in boiling water bath. You'll need to do a search to see how long.
I've gotten real lazy about canning anymore, and I just put them all in a big jar and keep in my refrigerator. They last indefinately that way.
Sometimes I even start with canned beets from the grocery store, and pour my pickling solution over the top.