What is the sign of mozzarella being old?
Mozzarella comes in different variations...  There are a few things to look and smell for to determine it they are "off".  Aged or drier varieties may have visible mold on the exterior. Pre-grated mozzarella can also exhibit visible mold as well as a sour or yeasty smell.  Brined varieties may have a musty, sour, or smelly-foot odor or an excessively cloudy brine. 

If you see visible mold on a drier variety, you can just cut it off and use the remaining cheese as normal as long as it doesn't smell.  If you plan on cooking or melting the cheese, it would likely be safe to eat, but may not taste normal.

Moist cheeses like mozzarella can harbor Listeria, E coli, brucella, and salmonella so if you are in doubt and not interested in a bout of food poisoning, it's best just to throw it out.

To keep brined varieties fresh longer keep it refrigerated and replace the brine daily, the replacement brine should taste salty like the ocean. Even so, it should really be eaten within 2-3 days.