Astronomy: What is the motion of the Moon such that an eclipse doesn't happen every month?

There is a New Moon every month simply when the whole of the side we can see is in shadow. However, to get a solar eclipse, it must be lined up more precisely with the Sun so that it blocks the Sun out. The orbit of the Moon is angled at about 5 degrees to the Earth's orbit so they only cross in two places, called the nodes. An eclipse can only happen when there is New Moon at a node.

If the two orbits were perfectly lined up in the same flat plane, then we'd get a solar eclipse (and a lunar eclipse) every month. But as they aren't, most months the Moon passes a bit above or below the Sun and there is no eclipse. Only when the Moon is at a node and it is New Moon at the same time can there be an eclipse, Which of course happens much less often.

A new moon is when the moon is between the Earth and the sun so its sunlit side faces away from us.

That happens every month.

A lunar eclipse is when the moon passes through Earth's shadow in space - that shadow is relatively small and the moon does not orbit the Earth in the exact same plane that the sun appears to move, so the moon is often either above or below Earth's shadow.

The moon's orbit is tilted by an angle of 5° with respect to the plane of the Earth. This moon's orbital however is in continuous movement. It takes 18 years for a full revolution. (Imagine a hula hoop toy moving around a kid's waist. Its movement resembles that of the moon around Earth). Now, when the spot where the moon passes in such a way it comes right between the sun and the Earth, a total eclipse of the sun is observed. These spots are two (so they happen every 9 nine years ) and are called "nodes".

I couldn't contribute here without parroting Carols answer.

The plane of the Moon's orbit is tilted at 5.12° degree angle to the plane of the Earth's orbit around the Sun, the ecliptic, is why. Eclipses happen only when the Moon gets close to where the two orbital planes intersect, called apsides. This is why eclipses often occur in pairs or triples 2 weeks apart.