Old dough is just what it sounds like. This is one of the quirkier pre-ferments. Usually this is dough that has matured to the point where it could have been baked but wasn't. Sometimes it's a day or two old.
The old dough will have developed complex, rich and often funky flavors. The old dough is added to fresh dough to give the bread these characteristics.
While many bakers make this on purpose, using accidents and overruns as old dough is a good way to minimize waste. So, if you have a bit of dough left over after loafing your dough, when a batch of dough collapses before you have a chance to bake it, if you don't get around to baking a batch of dough, or if you make dough for the purpose, you can let the dough mature a bit, save it, and use it as old dough.