Elongation
The elongation of the moon refers to the difference in location from the center of the moon to the center of the sun as observed from any position on Earth. At its shortest elongation there is a chance for a partial or total eclipse visible from that location on Earth. Nearby locations may also observe an eclipse.
Degrees
In terms of the elongation of the moon, degrees refer to the angles of distance between the moon and the sun in relation to the observer. Picture a triangle of moon, sun and observer; the degree refers to the perceived angle at the observer's corner of the other two. The elongation increases and decreases from that location as the rotation of the Earth, as well as the moon's path around the Earth, continue.
Phases
During a new moon the degree of elongation is zero, meaning the center of the moon and the center of the sun occupy the same visible position in the sky from that location. This phase offers the possibility of an eclipse depending on how high the moon and the sun are in relation to each other. During the first quarter of the moon phase the angle becomes 90 degrees in eastern elongation, meaning the sun is at a 90-degree angle to the observer. During the full moon phase the Earth is between the sun and the moon, giving it a 180-degree elongation and the possibility of a complete view of the moon from that location. During the third quarter the angle changes to 90 degrees in western elongation, positioning the moon at a right angle to the west of the sun from that position on the Earth.
Waxing and Waning
A "waxing" moon refers to the period in which the moon is moving upward in elongation, from a new moon to a quarter moon to a full moon, or when the moon is in the eastern quadrature. "Waning" refers to the period in which the moon is moving downward in elongation, through the western quadrature, from full moon to three-quarters moon to new moon. This process is further broken down into a waxing crescent, when the moon is between 1 and 89 degrees east, waxing gibbons, between 91 and 179 degrees east, waning gibbons, between 179 and 91 degrees west, and a waning crescent, between 89 and 1 degree west.