During the new moon phase, the moon is not visible in the sky. This occurs when the sun and moon line up like the hands of a clock at noon; the moon is too close to the position of the sun for people to see it from the Earth. A solar eclipse--when the moon blocks out the sun--can only occur during the new moon phase. Also, the Earth's tides are highest and lowest during the new moon.
During the half-moon phase, the sun illuminates only half of the moon's surface. Using the clock analogy, the half-moon shows twice a month, when the clock is at 3 o'clock and 9 o'clock. Tides are weakest during the half-moon phase.
The full moon phase shows the entire disc of the moon. This occurs when the sun, moon and Earth line up like the hands of a clock at 6 o'clock. A lunar eclipse--when the Earth's shadow blocks the reflected light of the moon--can only occur during the full moon phase.
The Earth's rotation has no impact whatsoever on moon phases. The phases of the moon depend entirely on the relative positions of the Earth, sun and moon. The Earth's rotation only affects which portions of the Earth are in sight of the moon at any given moment.