Q: What are the wispy white streamers that sometimes appear in vinegar?

Those white streamers are called "Mother of Vinegar".  They are clumps of the bacteria that turn alcohol into vinegar.  Generally, they are harmless and you can just pour the vinegar off the top, leaving the mother behind.  Sometimes, if left in the vinegar for a long time, they may cause a noticeable change in the flavor.  If you want to get rid of them, run the vinegar through a coffee filter into a clean bowl, rinse the container and pour the vinegar back in.

The bacteria may also start to digest the vinegar itself, making it weaker over time.

If you wish, you can use the mother to turn leftover wine or hard cider into vinegar.  Simply place the liquid into a sterilized container, stir in some mother of vinegar, cover with several layers of cheesecloth to keep out insects and debris and keep in a warm place until it sours.  Make sure to leave some headroom.  Also, since the process requires oxygen, a wider mouth container works better.

The ideal temperature is in the range of 60 to 85°F (15 to 30°C).  The total length of time it takes will depend on the temperature at which it is stored and access to oxygen, and may take up to about 4 weeks.  When you think it is ready, pour off a small amount and taste it.  If it has a nice clean vinegary taste, without any alcohol flavor, then it is done.  Strain it through a coffee filter and store in a clean bottle.  The mother can be recycled into another batch.

If there are any unpleasant tastes, then other bacteria have been at work and the vinegar is spoiled.  Discard it, and the mother, and start a fresh batch.

You can buy packages of the bacteria that are used to make vinegar at your local wine making supply store, or you can sometimes find mother of vinegar floating in bottles of unprocessed vinegar at health food stores.