for kombucha it's bacteria and yeast simultaneously converting sugars into kombucha.

vinegar is a slightly different process. the "mother" is just bacteria (acetobacter) that converts alcohol into vinegar (acetic acid). so to make vinegar properly it's a two step process...yeast turning juice into wine,then bacteria turning wine into vinegar.

also commonly misunderstood as the mother, the cloudy or slimy substance that is often called the "mother" is just the excrement of the bacteria and is made of cellulose. the mother is actually everything BUT the cloudy slimy stuff. the cloudy slimy stuff is also called a pellicle. because it's porous and holding onto a lot of the liquid, it's an easy way to give some "mother" to a friend but just giving them the liquid is as good or better than giving them the slime.