As the moon circles the Earth, the shape of the moon appears to change; this is because different amounts of the illuminated part of the moon are facing us. The shape varies from a full moon (when the Earth is between the sun and the moon) to a new moon (when the moon is between the sun and the Earth).
When two full moons occur in a single month, the second full moon is called a "Blue Moon." Another definition of the blue moon is the third full moon that occurs in a season of the year which has four full moons (usually each season has only three full moons.)
A crescent moon is part way between a half moon and a new moon, or between a new moon and a half moon.
A full moon appears as an entire circle in the sky. The full moon is given different names, depending on when it appears. For example, the "Harvest moon" is the full moon that appears nearest to the Autumnal Equinox, occurring in late September or early October. Some other full moon names (by month) include:
January Moon After Yule, Wolf Moon, or Old Moon
February Snow Moon or Hunger Moon
March Sap Moon, Crow Moon, or Lenten Moon
April Grass Moon or Egg Moon
May Milk Moon or Planting Moon
June Rose Moon, Flower Moon, or Strawberry Moon
July Thunder Moon or Hay Moon
August Grain Moon or Green Corn Moon
September Fruit Moon or Harvest Moon
October Harvest Moon or Hunter's Moon
November Hunter's Moon, Frosty Moon, or Beaver Moon
December Moon Before Yule or Long Night Moon.
A gibbous moon is between a full moon and a half moon, or between a half moon and a full moon.
A half moon looks like half a circle. It is sometimes called a quarter moon (this Moon has completed one quarter of an orbit around the Earth from either the full or new position and one quarter of the moon's surface is visible from Earth).
The new moon is the phase of the moon when the moon is not visible from Earth, because the side of the moon that is facing us is not being lit by the sun.
Moonrise from Earth
The moon rises and sets every day, appearing on the horizon just like the sun. The time depends on the phase of the moon. It rises about 30 to 70 minutes later each day than the previous day, so the moon is out during daytime as often is it's out at night. At the time of the new moon, the moon rises at about the same time the sun rises, and it sets at about the same time the sun sets. As the days go by (as it waxes to become a crescent moon, a half moon, and a gibbous moon, on the way to a full moon), the moon rises during daytime (after the sun rises), rising later each day, and it sets at nighttime, setting later and later each night. At the full moon, the times of moonrise and moonset have advanced so that the moon rises about the same time the sun sets, and the moon sets at about the same time the sun rises. As the moon wanes (becoming a half moon and a crescent moon, on the way to a new moon), the moon rises during the night, after sunset, rising later each night. It then sets in the daytime, after the sun rises. Eventually, the moon rises so late at night that it's actually rising around sunrise, and it's setting around sunset. That's when it's a new moon once again.
Crescent Moon - when we can see only a sliver of the moon's disk (the side of the moon facing us)
Full Moon - when the moon's disk is light because the Earth is between the sun and the moon
Gibbous Moon - when we can see roughly three-quarters of the moon's disk
Half Moon - (also called quarter moon) when we can see one half of the moon's disk (this is one-quarter of the entire moon's surface)
New Moon - when the moon's disk is dark (and invisible to us) because the moon is between the sun and the Earth
Quarter Moon - (also called half moon) when we can see one half of the moon's disk (this is one-quarter of the entire moon's surface)
Waning Moon - when the moon seems to be getting smaller, going from full to gibbous to half to crescent to new
Waxing Moon - when the moon seems to be getting bigger, going from new to crescent to half to gibbous to full