Why don't we get lunar and solar eclipses once every month on a full moon day and a new moon day respectively ?

In principle we should observe one lunar eclipse every month - on the full moon day, when the Earth is in between the Sun and the Moon.Similarly, we should observe one total solar eclipse every month - on a new moon day, when the Moon is in between the Sun and the Earth.

But this does not happen. Also lunar eclipse occurs more frequently that solar eclipse. The Earth revolves around the Sun in a Sun - Earth plane. The Moon revolves around the Earth in a Earth - Moon plane. These two planes, namely the Sun - Earth and the Earth - Moon plane do not lie in the same plane. The Earth - Moon plane is tilted by about 5 degrees with respect to the Sun - Earth plane.

Therefore, when there is a new Moon, the moon may not lie exactly in the Sun - Earth line. It may be a little below and its shadow may miss the Earth. Similarly, during a full moon, the moon is not exactly in line with the Sun and the Earth and hence misses the Earth's shadow. Lunar eclipse is observed more often because Earth's shadow is pretty large; so even if the Moon is not exactly in the Sun - Earth line, it may still be within the Earth's shadow for the lunar eclipse. For the solar eclipse, the geometry has to be pretty exact because the Moon is much smaller than the Earth. Solar eclipse occurs may be twice a year or even less.

Partial solar eclipses do occur frequently, but a total solar eclipse with the moon totally blocking the sun is a bit rare. Having the moon come exactly in between the Sun - Earth line is rare celestial coincidence and therefore the phenomena of total solar eclipse is a much awaited event.

There is another coincidence why a total solar eclipse occurs. The sun's diameter is 400 times the diameter of the moon. But it is also 400 times farther away from us than the moon is. Thus we can say that the size of the moon and the size of the sun as seen by us are same. To make things a bit more clear : hold a 50 paise coin and stand in front of a lighted bulb, about 1 meter away. Let the bulb be at your eye level. Watch the bulb and change the distance of the coin between your eye and the bulb (keep one eye closed). There will be one position where the coin will block the light from the bulb completely. The size of the coin is much smaller than the size of the bulb and yet it is able to block the bulb. The same phenomenon happens during the total solar eclipse.

It is known that our moon is slowly receding from us. So billions of years later this fortuitous coincidence will not be valid and there will be no total solar eclipses! But lunar eclipses may continue to take place.