Today is October 11 by the solar calendar (양력), but it is only September 9 by the lunar calendar. Since there are twenty-nine and a half days in a lunar month, that means it takes about 15 days for a new moon to wax to a full moon, and another 15 days for it to wane from a full moon to another new moon. Since today was the ninth day of this lunar month, that means the moon is waxing toward a full moon. In fact, since today is the ninth day of the lunar month, that means the moon is already more than half way toward a full moon.
Since there is both a waxing phase and a waning phase, that means there are actually two half moons (반달) in a lunar month. The half moon that occurs during the waxing phase is called the "first quarter moon," and the half moon that occurs in the waning phase is called the "last quarter moon." In Korean, the first quarter moon is called 상현달, and the last quarter moon is called 하현달.
Here are the four moon phases in Korean and English
신월(삭월): new moonAll right, but what are 초승달 and 그믐달?
상현달: first quarter moon
보름달(망월): full moon
하현달: last quarter moon
초승달 and 그믐달 are crescent moons. 초승달 is the crescent moon that forms at the beginning of a lunar month, and 그믐달 is the crescent moon that forms at the end of a lunar month. For example, on Day 2 of a lunar month one would see a 초승달, and on Day 28 of a lunar month, one would see a 그믐달. That means that the end of one crescent moon and the beginning of another is only a day or two apart.
Crescent moons bow inward. That means that a moon will remain a 초승달 until it waxes to its first half moon, which I mentioned was 상현달 in Korean, and "first quarter moon" in English. In English, we refer to a 초승달 as a "waxing crescent moon." Likewise, a 그믐달 will begin when the moon wanes past the second half moon mark, which I mentioned was 하현달 in Korean, and "last quarter moon" in English. After it passes that halfway mark, it appears to bow inward again and becomes a "waning crescent." It will remain a 그믐달 until it disappears into a new moon(신월/삭월), which is when the moon is completely dark.
So, if crescent moons bow inward, what do we call moons that bulge outward? In English, we call them "gibbous" moons. The moon between a first quarter moon and a full moon is called a "waxing gibbous moon," and the moon between a full moon and a last quarter moon is called a "waning gibbous moon." So what do Koreans call a waxing gibbous moon and a waning gibbous moon? Well, they call them 상현달 and 하현달, respectively.
Notice that 상현달 and 하현달 each have two meanings. 상현달 means "first quarter moon" and "waxing gibbous moon. 하현달 means "last quarter moon" and "waning gibbous moon." In other words, a 상현달 begins with and includes the first half moon and continues to the full moon, and 하현달 begins with the full moon and lasts until and includes the second half moon. Confusing? Well, maybe this list will help:
new moon: 신월(삭월)
waxing crescent: 초승달
first quarter moon: 상현달
waxing gibbous: 상현달
full moon: 보름달(망월)
waning gibbous: 하현달
last quarter moon: 하현달
waning crescent moon: 그믐달
(next) new moon: 신월(삭월)
Here is an interesting link that shows the phases of the moon and their Korean names:
Phases of the Moon