Orbit
The moon's orbit around the Earth is only very slightly elliptical; nonetheless, the moon is farther from Earth at some times than at others. At its nearest, the moon is 356,400 kilometers away from Earth; at its farthest, the moon is 406,700 kilometers away. The point of nearest approach is called perigee, while the point at which the moon is farthest from you is called apogee. The moon's angular size varies slightly between apogee and perigee.
Angular Size
Angular size or diameter is often measured in terms of arc-minutes, where an arc-min is 1/60 of a degree. At apogee, the moon's angular diameter is about 29.87 arc-min, while at perigee it's about 33.89 arc-min. On average, the angular diameter is roughly 31.1 arc-min or 0.5182 degrees. If the moon happens to reach perigee at the same time as the new moon occurs, this coincidence is sometimes called a "supermoon."
Approximation
The distance from Earth to the moon is much greater than the moon's actual diameter, so you can approximate the angular diameter with a simple mathematical equation: angular diameter = 57.3 * (1,738 km / distance to moon), where 1,738 km is the measured radius of the moon. If you take the average distance from Earth to the moon, which is about 384,000 km, and plug that into this equation, you'll obtain a result of 0.52 degrees -- the same number already mentioned. You can use this formula to estimate the angular size of the moon at other distances as well.
Eclipses
The sun's diameter is roughly 400 times that of the moon, but it's also 400 times farther from Earth than the moon is, so the angular size of the sun and the moon are roughly the same. This remarkable coincidence is a fortunate one for astronomers, because it makes possible the amazing phenomenon called a total solar eclipse, when the moon briefly passes directly in front of the sun, obscuring it temporarily.