Understanding the Moon's Phases
The new moon occurs when the moon is between the earth and the sun, when the lit-up part of the moon is facing away from us, so the moon is almost never visible on this day. The new moon rises and sets with the sun.
The waxing crescent is the period between the New Moon and the First Quarter. During this 6 or 7 day period, the moon appears as a crescent rising later and growing larger each day. The first day it rises shortly after the sun and sets shortly after the sun, and by the last day, it rises shortly before noon and sets shortly before midnight. The best time to view the waxing crescent is in the evening.
The first quarter marks the end of the first fourth of the moon's cycle. At this point the moon appears as a perfect half circle, as we can see half of the moon's lit-up side. The first quarter moon rises around sunset, making this the best time to view the moon. It sets around midnight.
Waxing gibbous is the the period between the first quarter and the full moon, and is also 6 or 7 days long. On the first night, the moon appears slightly more than half full, and by the last day, it is nearly full. The first night, the moon rises shortly after noon and sets shortly after midnight, and on the last night, it rises shortly before sunset and sets shortly before sunrise.
The full moon occurs when the moon is on the opposite side of the earth from the the sun, when the whole lit-up side is facing the earth, and it appears as a perfect circle of bright light. The full moon rises around sunset and sets around sunrise, and the best time to view it is around nine or ten o'clock.
The waning gibbous moon is the 6 or 7 day period between the full moon and the third quarter. It starts out slight less than full and ends up slightly more than half full. On the first night it rises shortly before midnight and sets shortly before noon. The best time to view it is between 9 at night and early in the morning, depending on the day.
On the night of the third quarter, the moon appears again as a half circle, but this time on the left side rather than the right. The third quarter moon rises around midnight and sets around noon, and the best time to see it is either late at night or early morning.
The waning crescent is the 6 or 7 day period between the third quarter and the new moon. Each night the crescent wanes thinner and thinner. The first day, it rises shortly after noon and sets shortly after midnight, and the last day, it rises shortly before sunrise and sets shortly before sunset. The best time to view it is late night to early morning.
A lunar eclipse occurs when the earth's shadow falls on the moon. Because this only happens when the earth is directly between the moon and the sun, the moon is always full on the night of an eclipse. There is not an eclipse on every full moon because most full moons are not exactly on the opposite side of the earth from the sun, but slightly off, so that it doesn't fall in the earth's shadow.
Light reflected from the earth, "earthshine", causes the moon to appear red during a lunar eclipse.
A solar eclipse occurs when the moon's shadow falls on the earth. This only happens when the moon is directly between the sun and the earth, so it only occurs on the day of the new moon. Again, this doesn't happen every new moon because most new moons aren't exactly between the sun and the moon, but slightly off.