Let's look at the diagram below - the phases of the Moon are entirely due to the geometrical relationship between the Earth, Moon, and Sun. By the way, contrary to popular misconception, the phases of the Moon ARE NOT due to any shadow from the Earth on the Moon (lunar eclipses are, but that's a different story).
Back to the diagram of the Moon orbiting the Earth.
Similar to the Earth, the side of the Moon facing the Sun is also lit and the side facing away is unlit. Day and night on the Moon occurs for the same reason it does on the Earth. There is no permanently "dark side of the Moon" despite Pink Floyd album names. The only different is that a "day" on the Moon is much longer than a day on the Earth because the Moon rotates on its axis much more slowly. More on this later.
Anyway, as you can see from the diagram above, when the Moon lies between the Earth and the Sun, the side facing the Sun is lit and the side facing the Earth is unlit. Also note that such a Moon is high in the sky around noon for the daytime Earth (the side of the Earth that's lit). Therefore we simply don't see the Moon in the sky. It's the New Moon phase.
Conversely, when the Moon is on the opposite side of the Earth from the Sun, the side facing the Earth is completely lit and it's high in the night sky around midnight. Therefore we see it as a Full Moon.
The Moon orbits the Earth in a counterclockwise fashion (as seen from above the North Pole) just as the Earth rotates counterclockwise. So you can see the 1st Quarter Moon is high in the sky around sunset and from the Earth, you would see the right-hand side of the Moon as lit. Similarly, the 3rd or Last Quarter Moon would be seen around sunrise and you would see the left-hand side lit (unless you're upside-down in the Southern Hemisphere).
Hopefully, you can also visualize why you see waxing or waning crescent or gibbous moons. From the perspective of being on the Earth, you will see either less than half the Moon lit (crescent phase) or more than half the Moon lit (gibbous phase). Note that waxing crescents would be seen in the afternoon and waning crescents in the morning (before noon). Waxing gibbous is after sunset and waning gibbous before sunrise. In other words, don't bother looking for a crescent moon high in the sky in the middle of the night. It will either be setting in the evening after sunset (with the right side lit) or rising in the early morning hours before sunrise (left side lit).
Make sense? Next we'll talk about how long it takes the Moon to orbit around the Earth.