It is a well-known fact that the moon does not have light of its own and illuminates when it receives light from the Sun. The light of the Sun reaches the moon and bounces back and it appears to us that it is the light of the moon.
As the Earth revolves around Sun and the moon revolves around Earth, a time comes when the three bodies are in a line and Earth is in the middle of the Sun and moon. Thus, the moon moves into the shadow of Earth, and the light from Sun does not reach the moon.
Due to this, the moon does not reflect light, and we are unable to see the moon. This phenomenon is called the Lunar eclipse.
Full moon day:
As the moon does not emit its own light, and the light coming from it is actually sunlight. So, due to the moon orbiting around the Earth, the sunlight hits different parts of the moon, and only that part is visible to us. So, it looks like the moon is changing its shape daily.
A full moon occurs when sunlight falls on the whole surface of the moon.
On a lunar eclipse, since the Sun and moon are in the same line, the sunlight must fall on its whole surface. Therefore, it is a full moon day on every lunar eclipse. But this does not mean that lunar eclipse will occur on every full moon because it is not necessary that whenever the sunlight falls on the whole moon, the Earth will also be in between them.
Hence, lunar eclipse always occurs on a full moon day.