The Moon, like every other planet and satellite in the solar system, shines not with its own light but with the reflected light of the Sun. This means that at any point in time one half of the Moon is lit while the other half is in darkness. As the Moon revolves around the Earth, over a period of about 29 days, a varying proportion of the illuminated half can be seen from the Earth. The boundary between the light and dark halves, called the terminator, migrates from east to west across the lunar surface and, because the Moon is spherical, creates the familiar phases: crescent, full, etc.
The Moon rises and sets in roughly the same directions as the Sun, but about 49 minutes later each night than the night before. The cycle of phases, called a lunation or synodic month, begins with a new moon, when the Moon passes close to the Sun and so cannot be seen against its overwhelming glare. This is the time when solar eclipses can occur, but for reasons described in another article they usually do not.
A couple of evenings later the Moon has moved away from the Sun. It sets a little after sunset, and so can be briefly observed as a thin crescent low in the west. The phase of the Moon facing the Earth is always the exact opposite of the phase of the Earth facing the Moon, so although very little of the sunlit portion of the Moon is visible from the Earth there is quite a lot of earthshine on the dark portion. Sometimes this is enough to faintly reveal the Moon's whole disc, a phenomenon known as "the old moon in the new moon's arms".
Over successive nights it becomes easier to observe as it sets progressively later and as its phase becomes larger (called waxing). About a week after the new moon the phase has reached a semicircle. Despite the shape, this a called the first quarter, because it marks one quarter of a lunation. This is the phase most often seen during the daytime, as it rises around noon, sets at about midnight, and is highest around 6pm. In fact, over a month, the Moon is in the sky for as many hours during the day as during the night. However it is only at this time, when it begins to spend more than half of its time as a nighttime object, that it has become sufficiently well illuminated to show up in the bright daytime skies.
As the waxing continues, the next phase is called gibbous - a disc with just a crescent missing from it. Then, some 14 days after the new moon, we see a full moon - the Sun and the Moon are on opposite sides of the Earth, and when one sets the other rises. This is the period when lunar eclipses can happen, but for the same reasons as for solar eclipses they remain unusual.
From this point onwards for the remainder of the lunation the Moon's phase becomes thinner again, called waning. It passes through gibbous phase, then roughly three weeks after new, it becomes a half circle again, the mirror of the first quarter, called (for obvious reasons) the last quarter. Again this is a phase often seen in the daytime, but during the morning as it rises at about midnight and sets around noon.
The last phase is a crescent again, ending with another chance to see the effect of earthshine (this time called "the new moon in the old moon's arms"). Then, about 29 days after the last new moon, another takes place, with the Sun and Moon rising and setting at much the same times, and the whole process repeats.