All you have to do is find the moon and glance at it briefly, a few times over a period of a month,
to see that no single phase of the moon is "always" seen. It changes constantly.
No, the visibility of the moon depends on its phase and its position in the sky. The moon is not always visible because it goes through different phases, such as new moon when it is not visible at all and full moon when it is fully illuminated and visible all night.
The waning Moon phase in which only one-half of the lighted side of the Moon can be seen from Earth is called the Third Quarter Moon. It occurs when the Moon has completed three-quarters of its 29.5-day cycle.
That would be Third Quarter.
Full moon, or within one day of the full.
No, a lunar eclipse is not considered a moon phase. A lunar eclipse occurs when the Earth passes between the Sun and the Moon, casting a shadow on the Moon, while moon phases refer to the different illuminated portions of the Moon as seen from Earth during its orbit.
After the waxing crescent phase, the Moon enters the first quarter phase. During this phase, half of the Moon's surface is illuminated as seen from Earth, and it appears as a half-circle. The first quarter occurs roughly one week after the new moon, as the Moon continues to wax towards the full moon phase.
That's "First Quarter", roughly 7.5 days after New Moon.
The moon displays a complete cycle of all its phases every 29.53 days.
The Waxing Moon phase occurs when we can see the right half of the Moon illuminated from Earth as it moves towards a full moon. This phase follows the New Moon and precedes the Full Moon in the lunar cycle.
The moon's surface is always half illuminated. We on Earth see 0 to 100% illumination of the side that faces use, depending on what phase the moon is at. Quarter phases, first and last, are half illuminated when seen from Earth.
One week from now, the moon will likely be in the first quarter phase, where half of the moon is illuminated. The new moon phase is followed by the waxing crescent phase and then the first quarter phase in the lunar cycle.
Approximately one week after the new moon phase, the waxing crescent moon phase occurs. This phase is characterised by a small sliver of the moon becoming visible on the right side.