The moon clearly exhibits a light and dark portion during most of the month except during full moon. What causes the dark portion of any phase to change in size, that is, become either more or less black?

While rotating around Earth the Moon also rotates around it's axis. These two rotations have almost same duration (27.3 and 27 days respectively). This is the reason why we always see the same side of Moon.

The dark part of a Moon is the part where there is night on the Moon, meaning there is no sunlight visible. This dark part depends on the Moon's position relative to Earth and Sun. The change in size is due to part of Moon-day. Dark part of Moon has night, bright part has day and the boundary has either sunrise or sunset.