Eye dominance is the tendency to prefer one eye over the other. If you're right-handed, your right eye is usually dominant.
However, this doesn't always work this way. Both hemispheres of the brain control both eyes, but one will always assume responsibility. Dominance appears to change depending on the direction of gaze due to changes in the image size on the retina.
Lucky archers who dominate the right eye and right hand (or the left eye and left hand) actually have the opportunity to keep both eyes open while shooting.
Not only does this allow more light to be “let in,” you also have twice the visual information being fed into the brain to aid reaction and aiming.
The dominant eye always takes control.