The "Reincarnations" of the Moon
The so-called age of the moon in a particular phase is the number of days past the instant of an exact new moon. So we speak of the moon as if the old moon dies at the end of each cycle of phases and a brand new moon is born again at the instant of new moon.
The Real Age of the Moon
No doubt many earlier cultures did indeed think of the waxing and waning of the moon in exactly this way, with an actual new moon being born at the start of each cycle of phases, but it's nonsense of course. As a physical object, the age of the moon is of approximately the same age as the sun, earth, and solar system as a whole - namely about 4.6 billion years.
The Origin of the Phases
The changing shapes that we see for the moon over the course of the month arise merely because the moon is a dark spherical object illuminated on one side by the sun. We can only see the illuminated portion of the moon. So the apparent shape of the moon in the sky is determined by how much of the illuminated hemisphere is turned towards the earth. That in turn depends on the apparent angle between the sun and moon in the sky. Go here for a diagram and discussion of how the phases of the moon originate.
The Ages of the Phases
Since a complete cycle of phases takes approximately 29.5 days - the synodic month - the age of the moon in any phase will always be less than 30 days. First quarter will correspond to an age of about 7.4 days, the full moon - about 14.8 days, and third quarter - about 22.1 days. The moon will be virtually invisible in the sky from a day or two before new till a day or two after. So you won't see the old moon much after an age of maybe 28 days or so, and you won't see the moon reborn until it has an age of something more than a day.
