Our Moon is relatively close to our planet Earth and some details on its surface are visible even with a naked eye. When the Moon moves across the sky, you can notice that the boundary between its illuminated and non-illuminated part moves from the right to the left. This boundary is called the TERMINATOR. When the Moon is exactly between the Earth and Sun, we do not see it - we speak about the New Moon. Then, the MORNING terminator starts to move from the right to the left, i.e. the Sun RISES on the Moon's surface exactly on the morning terminator. If you were standing on the Moon's surface exactly on the morning terminator, the Sun would rise for you naturally on your EASTERN horizon (this is the definition of LOCAL EAST on any planet or moon). However, if you stand on the Earth's surface and look towards the Moon, then you see its illuminated part to the right, i.e. to the LOCAL WEST on the Earth. Hence, the right boundary of the Moon is towards Moon's LOCAL EAST, and its left boundary is towards its LOCAL WEST. So, please, remember: local East and West on the Moon are towards Earth's local West and East, respectively, i.e. they are INTERCHANGED with respect to the Earth. There is no problem with the Moon's local North and South, which are towards the top and bottom of the Moon's disk, respectively.
When the morning terminator halves the Moon's disk, we have the First Quarter. Approximately after another 7 days the morning terminator reaches the left boundary of Moon's disk (i.e. the Moon's west) and we have the Full Moon. Then, on the right (locally eastern) Moon's boundary, the EVENING terminator appears and moves again from the right (Moon's east) to the left (Moon's west). If you were standing on the Moon's surface on the evening terminator, you would see the Sun SETTING DOWN on your local western horizon (this is the natural definition of the local west). When the evening terminator halves the Moon's disk, we speak about the Last Quarter. Then, after another 7 days, the evening terminator reaches the left (Moon's western) boundary of the Moon's disk, the Moon is exactly between the Earth and Sun, and we have again the New Moon.
Please notice that the full cycle of the Moon's phases takes about 29 full days (i.e. 29 x 24 hours) as measured on Earth. For a given, fixed place on the Moon, approximately one half of this time (14.5 full days on Earth) the Sun is over the local horizon (= one "day" on the Moon), and another half of this time (14.5 full days on Earth) the Sun is under the local horizon (= one "night" on Moon).
This never-ending play between the darkness and light, the Moon's day and night, is the easiest observation on the Moon that can be made by naked eye.