Mother of Vinegar is a bacterial starter culture used to make vinegar.
Mère de vinaigre is a slimy, gummy substance, that you wouldn’t normally want to pitch into something you are later going to consume.
When added to the liquid to be made into vinegar, it causes the fermentation that creates the acetic acid and makes vinegar — acetic acid being what gives vinegar its sour taste.
Once the liquid has turned to vinegar, the mère will have used up whatever sustained it and it has to be either discarded, or transferred to a new batch of soon-to-be vinegar to keep it alive, just as you would a starter yeast for baking.
Often, it’s called by its French name, “mère de vinaigre”.