The weight of a body on the Earth is not in fact the force exerted on it by the Earth, and the weight of a body on the Moon is not the force exerted on it by the Moon. The weight of a body is in fact the total gravitational force exerted on it by all the other bodies in the Universe, though of course when the body is on the Earth, all other force acting on the body will be negligible compared to that acting on it by the Earth.
If we ignore that technical point and consider the weight of a body on the Earth to be due only to the gravitational force acting on it by the Earth, then it's weight is given by
We know that the acceleration of a body towards the Earth is due to the Force of gravity so we can find the acceleration using
The weight of a body on the Earth is not constant, for three reasons.
The Earth is not precisely spherical. Technically it is an oblate spheroid, slightly flattened at the poles. The acceleration due to gravity will be slightly greater at the pole for this reason.
The distribution of mass over the Earth is not regular. The Earth's surface has denser parts – ocean crust, and less dense parts – continental crust. Water is less dense than crust and occupies two thirds of the Earth's surface. There are mass concentrations in mountainous regions.
The Earth is rotating. Bodies on the Earth's surface experience centrifugal forces to some degree. The centrifugal force is maximum at the equator and decreases towards the poles. The difference is about 0.3%.
We are fortunate for gravity to have the value it does. For life to have evolved on Earth, the Earth needed an atmosphere. If the force of gravity on Earth were less than it is, the Earth might not have been able to hold onto it's atmosphere for 4.5 billion years – it could have been lost to space in the same way as Mar's atmosphere, and the Earth would now be a dead planet.