Very interesting question. A lot of people confuse the phases of the moon with solar and lunar eclipses. The phases of the moon are strictly due to how much light is illuminating the face of the moon that we can see at a given time. An eclipse occurs when the sun, moon, and earth are in perfect alignment. When the moon is between the sun and earth, you get a solar eclipse - because the moon is so much closer to us than the sun, the moon appears as big as the sun and can block out the sun. When the moon is behind the earth, directly in the earth's shadow, you get a lunar eclipse (like a dimming of the moon). Because the earth's orbit and the moon's orbit are on different planes, they do not always perfectly line up with the sun. That is why you can have a new moon without having a solar eclipse, and a full moon without having a lunar eclipse.
Here is a link to a web page that has some diagrams to help visualize this stuff. eclipse
In particular, look at the first diagram of the phases of the moon, and the diagrams about 3/4 of the way down the page that shows the moon's orbital plane and the earth's orbital plane (called the ecliptic since the orbital path is an eclipse). You can see that there will be at least two eclipses during the year - when the two orbital planes intersect. Because both the moon and earth actually wobble like a top spinning on the ground, their orbits change slightly and there can be more than two intersections of the orbital planes. This also makes it so that eclipses do not always occur on the same date every year.
The orbits of the moon and the Earth are not in perfect alignment; there is an angle between the plane containing the Earth's orbit and the plane containing the moon's orbit. This means that the moon is only occasionally exactly on the far side of the Earth from the sun (a lunar eclipse), and similarly only occasionally exactly between the Earth and the sun (a solar eclipse). Usually, the sun and the moon just barely miss each-other in the sky from the prospective of a person standing on Earth.
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