So. To describe the phases, let’s start at the beginning: New Moon.
New Moon happens when the Sun, Moon, and Earth are all more or less in a line. The Moon’s orbit is actually tipped a bit to the Earth’s, so sometimes new Moon happens when the Moon is “below” the Sun, or “above” it. But at some point in its orbit, at some point in the month, it appears to be as close to the Sun as it can.
What does this look like from Earth? The Moon is between the Earth and the Sun, so from our perspective we only see the dark half, the unilluminated half, of the Moon. The other side, the far side, of the Moon is lit, but we can’t see it. It makes sense then to call this the beginning of the Moon’s cycle, hence the term “New Moon”.
Now think about this for a sec: Because the Moon is near the Sun in the sky, it travels across the sky with the Sun. It’s up during the day! You can only see it from the part of the Earth that’s lit, which is when it’s daytime. It’s a very common misconception that the Moon is only up at night, but it’s up during the day literally just as often. At New Moon, the Moon stays near the Sun, so it rises at sunrise, and sets at sunset. This makes it extremely difficult to see; it is, after all, sitting next to the brightest object in the sky, and only a little bit of it is lit from our perspective.
But not for long. Because the Moon is orbiting the Earth, after a couple of days it’s moved a bit to the east. Now we’re seeing it along a slight angle, and we can see a little bit of the illuminated half of the Moon on its side toward the Sun.The terminator, the day/night line, appears curved around the Moon, so what we see is
a thin illuminated crescent Moon. At this point the crescent is still very thin, with
the horns of the crescent pointing away from the Sun. Note that the Moon is still pretty close to the Sun in the sky, just a bit to the east, rising maybe an hour or two after sunrise. But this means it’s up all day, and then sets after the Sun does. This is the best time to see the crescent Moon, when the Sun has already set and the sky starts to get dark. The Moon will be low over the western horizon, and it will set soon after the Sun does. Let’s wait a couple of days. OK, now the Moon has moved a bit more in its orbit around the Earth, and is farther from the Sun in the sky. We see a little more of the illuminated part, and the crescent is wider. Since it’s getting thicker, we say this is a “waxing crescent” Moon; waxing means growing or getting bigger. It’s also well away from the Sun now, so it’s easier to spot, even during the day before sunset. Seven or so days after new Moon we get to our first milestone: the Moon is now one-quarter of the way around its orbit. It’s 90° away from the Sun in the sky, which means we’re looking straight down on the terminator, the Moon’s day/night line. It cuts right down the middle of the visible face of the Moon, so it’s half lit, with the sunward side of the Moon visible and the other side dark.
Confusingly, this phase is properly named “first quarter” because the Moon is ¼
of the way through its cycle, ¼ of the way through its orbit around the Earth, even though it looks half full. So it’s not really the half-full moon -- astronomers prefer “first quarter,” so if you want to sound all astronomery, then you should call it that.
But time marches on. The Moon continues on its gravitational dance with Earth, swinging around its orbit. Now more than half full, we say its shape is “gibbous”, which means swollen or convex. Since it’s getting wider, this is actually the waxing gibbous phase of the Moon. It rises in the late afternoon, and is up most of the night. Our next big step comes two weeks after new Moon, when it’s moved halfway through its orbit. It is now opposite the Sun in the sky, 180° around. The Earth is between the Moon and Sun, so we’re looking at the fully-illuminated half of the Moon. This is the full Moon. Because it’s opposite the Sun, it rises at sunset and sets and sunrise; it’s up all night shining down on the Earth. But again, wait a couple of days and things change. When the Moon is full it’s 180° around the sky from the Sun, so as it continues to move around the Earth in a circle the distance between it and the Sun is now starting to decrease, even as it continues on in the same direction. As before, it keeps rising and setting later, but now it rises after sunset,
and sets after sunrise. If you get up early in the morning as the Sun is just rising in
the east, you’ll see the nearly-but-not-quite full Moon setting in the west.
Not only that, but we’re about to go through all the phases again, but in reverse order. A few days after full Moon the lit side is shrinking. It’s in the waning, or shrinking, gibbous phase. Then, three weeks or so after new Moon, and a week after full, the Moon is once again half lit, the terminator splitting the Moon’s face in two even halves. This is the “third quarter” Moon, because the Moon is ¾ of the way through its cycle. It’s a lot like the first quarter, but the side that was lit is now dark, and vice versa. It’s 270° around the sky from the Sun. It rises at midnight and sets at noon. A few days later and the Moon is a crescent again, getting thinner. It’s now a “waning crescent.” It rises just a couple of hours before sunrise, and sets a couple of hours before sunset. Then, finally, we’re back where we started. One month after new Moon, the Moon has traveled 360° around the sky, and is once again as close to the Sun as it can get. It’s new Moon, and the cycle starts up again as it has for time immemorial.