To begin with, the word for Moon is Japanese is tsuki 月. That's the same kanji as used for the word "month," gatsu 月, and make sure you remember it because it's important. Its Moon phases, gessou 月相, are:
- New Moon
- Waxing crescent Moon
or jougen 上弦
- Full Moon
- Waning crescent Moon
or kagen 下弦
For the waning and waxing Moons, the word gengetsu 弦月 can be used for either. Since the first quarter and the third quarter are also called "half Moon," you can use he word hangetsu 半月 to talk about. The han 半 there means literally "half."
Another thing about these two phases is that jou 上 and ka 下 are written with the kanji for "up" and "down" respectively. So a jougengetsu 上弦月 is a half-Moon, gengetsu 弦月, that is raising, increasing, while a kagengetsu 下弦月 is lowering, decreasing.
The shin 新 in shingetsu 新月 means "new," like in the adjective "new," atarashii 新しい. On the other side, the man 満 in mangetsu 満月 means "full" and is found in words like "satisfied," manzoku 満足.