When is there zero net force on an object?

The formal answer is: 
There is zero net force on an object when either there is no force on it at all, or else the strengths and directions of all forces acting on it cancel out and add up to zero.  When that's true, we say that all the forces on the object are "balanced".

How you can tell:
There is zero net force on an object when neither its speed nor its direction of motion is changing.  That means it's either at rest, or moving with constant speed in a straight line. 

(It doesn't take any force to keep an object moving in a straight line at constant speed, only to change speed or direction.  Either of those changes is called "acceleration".)