Weightlessness is the complete or near complete absence of the sensation of weight. It is a situation in which the gravitational force is 0. We feel weight because the ground exerts an equal and opposite force on our body after our body exerts a force on the ground due to gravitational attraction.
It occurs when there is no force of support on your body.
Weightlessness at the center of a planet:
The net gravitational force due to a spherically symmetrical planet is zero at the center. This is clear because of symmetry, and also from Newton's shell theorem which states that the net gravitational force due to a spherically symmetric shell, e.g., a hollow ball, is zero anywhere inside the hollow space. Thus the material at the center is weightless.