What is this box connected to my electric water heater?
Mike - that looks like a power saver box installed by your utility company.

The utility can regulate the HW Heater on and off. Thus saving you energy as the theory goes (that's the theory). 

My theory is if I want it low , I will set it there and not raise it high - thus I save energy and have hot water when and how I like it, not luke warm water.