Question

What is it called when you connect an object to earth with a conductor

Answer 1

This is called "grounding". Grounding gives electrical current a direction into the ground as opposed to another part of the system that you may want to have spared current.

Answer 2

In the US, most electricians and engineers call it "grounding" the object.
In many other places in the world, it may be called "earthing" the object.