Full Moon and New Moon phases occur every month.
The moon passes between the earth and the sun about once every 29.5 (new moon). In addition, the earth is directly between the sun and moon once every 29.5 days(full moon), about two weeks before and after every new moon. These alignments can be seen in the simulation below which uses svg and css code adapted from a solar system simulation authored by Jonimar Marques Policarpo.
In the simulation below, the red line shows the direction from the earth to the sun. When the moon passes between the earth and sun, a new moon occurs as the moon crosses the red line. On the other hand, when the moon passes on the opposite side of the earth from the sun, a full moon occurs when the moon crosses the red line. Those alignments (full moon and new moon) occur about 13 to 14 times per year each. However, only a fraction of all full moons and new moons result in eclipses. In fact, there can be no more than seven solar and lunar eclipses out of the 27+ full moons and new moons that occur each year. Occasionally, there are only four eclipses per calendar year. Let's find out why eclipses are so rare.
Why not every new moon or full moon results in an eclipse
The moon's orbit is tilted with respect to the earth's orbit
The earth's orbit is contained in a plane termed the ecliptic plane. The plane of the moon's orbit is inclined at about five degrees from the ecliptic plane. As a result, the moon can be well above or below the plane of the ecliptic at the time of a new or full moon. When a new moon occurs with the moon well away from the ecliptic, the shadow of the new moon will completely miss the earth. Alternatively, when a full moon occurs with the moon far from the ecliptic plane, the moon will pass above or below the earth's shadow. In either of those cases, no eclipse can occur.
Eclipses occur when the full moon or new moon occurs near a node
The moon's orbit intersects the ecliptic at two points on opposite sides of the earth. The scientific term for the points of intersection is "nodes". When the moon passes from south to north through the ecliptic, the node is an ascending node. On the other hand, when the moon passes from north to south through the ecliptic the node is an descending node. When the moon is at or near either node, the moon is very close to the plane of the ecliptic. Accordingly, when a new moon or a full node occurs near a node, an eclipse will occur. Near means a separation of 17 degrees or less from either node. In the simulation, the yellow bands along the moon's orbit indicate two 34 degree regions of the moon's orbit in which an eclipse is possible.
The math for numerophiles
Fraction of new and full moons that produce eclipses
As discussed above, there are two 34 degree bands out of the 360 degrees of the moon's orbit in which a new or full moon will produce an eclipse. Accordingly, about one-fifth (68/360) of the 27+ earth, sun, and moon syzygy will produce an eclipse. There were 1223 new moons during the period between January 1, 1901 and January 1, 2000. (68/360) * 1223 is about 231, a number in close agreement with the 228 solar eclipses and the 229 lunar eclipses which actually occurred duing the 20th century.
Eclipse Seasons
The orbital period of the moon is 27.3216 days. Concurrently with the moon revolving around the earth, the earth revolves around the sun once every 365.2422 days. The moon's orbit around the earth is not uniform; the moon's orbit is an ellipse rather than a circle. However, the combination of the earth's motion and the moon's motion produces a new moon every 29.5305 days on average, with a full moon occurring half a rotation (14.77 days on average) after each new moon.
1/(1/27.3216 - 1/365.2422) = 29.5305 days
In addition, the moon's orbit rotates slowly around the earth with a period of 18.6 years. The simulation depicts this change showing that the yellow arcs rotate clockwise slowly around the earth. This orbital rotation is in the opposite direction of the revolutions of the moon and the earth. Without the rotation of the nodes, there would be an alignment of the new moon and full moons with the nodes every 1/2 year. However, with the rotation of the moon's orbit, which amounts to about 19.35 degrees a year, the alignment frequency boecomes slightly less than one half year.
In order to calculate the period between eclipse seasons, we note that the earth revolves around the sun at exactly 360 degrees per year. Concurrently with the earth revolving, the nodes of the moon's orbit revolve around the earth once every 18.6 years in an opposite direction. So the total revolution is (360 + 360/18.6 degrees) per year or 379.35 degrees per year. The time to rotate 180 degrees is then (180 degrees / 379.35 degrees per year) * 365.2422 days per year which produces a period of 173.30 days.
Frequency of eclipses
As a result of the various periodicities described above, the moon, earth, and sun are in a proper alignment to produce eclipses every 173.3 days. The term for these periods of alignment is eclipse seasons. An eclipse season is a month long (34.5 day) period in which two or three eclipses will occur. The eclipses in a season will alternate between solar and lunar eclipses. In a calendar year, there will always be at least two eclipse seasons, but if an eclipse season starts early enough in January, part of a third season can occur in a calendar year. Consequently, the number of eclipses that occur during a calendar year can be as small as four and as large as seven.
This article explains the frequency of eclipses. A great tool for identifying exactly when an eclipse will occur and where on earth is the best place to see it is the Eclipse Planner app for your smartphone or tablet.