Is there a technical word for the circuit that a relay turns on
The Load.

The circuit that is 'powered' or 'controlled' by another component or circuit is often referred to simply as the load of the circuit.

It is referred to as a load as it 'loads down' the power supply, just as you would be loaded down should you be carrying something - the greater the load, the slower you will move, much like in a circuit, the bigger the load, the 'slower' current will flow.
If there was too little load a person would be able to run quicker and without common sense / ability to feel pain your body would start to break down, much like a power supply could burn out if too much power is drawn from it!