The entire illuminated portion of the moon is on the back side of the moon, the half that we cannot see. Gibbous refers to the shape, which is larger than the semicircle shape of the Moon at First Quarter, but smaller than a full circle.
This traditional definition of New Moon is still in use in some cultures, defining the beginning of the months for instance in the Islamic calendar.
A waxing crescent moon is when the Moon looks like crescent and the crescent increases "waxes" in size from one day to the next. The earth, of course, is at the center of the diagram. This figure is called the sidereal period or orbital period. Hence, complex equations are used to determine the exact position and phase of the moon at any given point in time.
By observing the shadow of the Earth across the face of the Moon during a lunar eclipse, Aristotle reckoned that the Earth was also a sphere. When moonrise precedes midnight one night, the next moonrise will follow midnight on the next night so too with moonset.
We call it "first quarter" because the moon has traveled about a quarter of the way around Earth since the new moon. Nothing on Earth was quite like this strange brilliant object arcing through the night sky.
Waxing means that it is getting bigger. The sun's light is now shining on the other half of the visible face of the moon. The Greeks were among the first to take a scientific look at the Moon and her phases.
So the basic explanation is that the lunar phases are created by changing angles relative positions of the earth, the moon and the sun, as the moon orbits the earth.