Light on the right means the moon is waxing (going towards a full moon), while light on the left means the moon is waning (going from a full moon).
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When the right half of the moon is illuminated it would be called first quarter. (That's when you see the Moon from the northern hemisphere.)
Waxing moving towards a full moon.
Half of the Moon, just like Earth and the other planets, is more or less illuminated all the time. The half that is illuminated is the half that faces the Sun. The illuminated half continually changes as the Moon orbits and revolves. The exception is when the Earth gets between the Moon and the Sun, shading the Moon. (an eclipse).
Yes, a gibbous moon appears more than half illuminated to people on Earth. It is illuminated between half and full, but not fully illuminated like a full moon.
If you live in the Northern Hemisphere, the left side of the moon is illuminated. If you live in the Southern Hemisphere, the right side of the moon is illuminated. Regardless of hemisphere, the western half of the moon is illuminated during the waning moon phases.