Our Moon's shape doesn't really change — it only appears that way! The “amount” of Moon that we see as we look from Earth changes in a cycle that repeats about once a month (29.5 days). The relative positions of our Sun, Earth, and Moon, cause these changes.
As our Moon orbits around Earth, the side facing the Sun is always illuminated, just like Earth's daylight side is illuminated by the Sun.
What we see from Earth, however, is a different story. Starting with the dark new Moon, we see the light part of the Moon “grow” from a sliver to a half to a full Moon — and then the illuminated part decreases, becoming thinner until there is no visible Moon in the sky and we are at the new Moon part of the cycle again.
We have a “new Moon” when our Moon's orbit around Earth moves it between Earth and the Sun. From Earth, the Moon's surface looks dark because the illuminated side is facing away from Earth. As our Moon continues its orbit counterclockwise around Earth (viewed from above the north pole), more and more of the illuminated part of the Moon becomes visible to us, until it reaches the “full Moon” stage. A full Moon occurs when the Moon has moved in its orbit so that Earth is “between” the Moon and the Sun.
Between the new and full Moon, the amount of Moon we see grows — or waxes from its right side toward its left side. As it passes the full Moon stage, the amount of illumination decreases — or wanes — from right to left. Finally, the Moon returns to its position between the Earth and the Sun, and on Earth we observe the new Moon again.
In the southern hemisphere, illumination of the Moon increases from the left to the right side in the waxing phase and the dark part increases in coverage from left to right in the waning phase, which is opposite of the northern hemisphere. No matter where on Earth an observer is, however, the phases of the Moon occur at the same time.