What is the name of the force that the earth exerts on objects?

The force of attraction between any two bodies is directly proportional to the product of their masses and inversely proportional to the square of their distance, according to Newton's universal law of gravitation.

All bodies on Earth have a weight, or downward force of gravity, proportional to their mass, which is exerted on them by the Earth's mass. The acceleration that gravity imparts to freely falling objects is used to quantify gravity.

Gravity is the gravitational attraction force imparted by the earth on any object.

Hence, gravitational force is the name of the force that the earth exerts on objects.