What is gravity?

What Newton came up with.

Over three centuries ago, Isaac Newton invented a new kind of mathematics called calculus so that he could model motion in the natural world using mathematics. Calculus is about measuring change and so calculus became a vital tool in describing the motions of simple objects.
Newton was able to make a mathematical model that encompassed both objects falling because of gravity on Earth, and the motion of heavenly bodies in the skies.
Newton decided that the force of gravity on Earth was the same force that organized the motions of the moon around the Earth and the Earth and all the planets around the sun. He invented a formalism and developed mathematical formulas for calculating the size of the gravitational force both on Earth and in outer space.
One of the important formulas in Newton's model is his law for calculating the force of gravity between two objects 1 and 2 with mass m₁ and m₂, which are separated by a distance R:

F₁₂ = F₂₁ = G m₁ m₂ / R²

The constant G is a number that occurs in Nature, like the speed of light c. The constant G is called Newton's gravitational constant.
Newton's law of gravity winds up describing the observed motions of the planets extremely well. Another thing it models quite well is the way the gravitational force felt on the surface of a planet depends on the size and mass of the planet. For example, comparing the gravitational force at the surface of the Earth vs. the moon, we get

F_moon/F_earth = (M_moon/M_earth) (R_earth²/R_moon²)

which is about 1/6, and the astronauts who walked on the moon felt it, too. You can see how much lighter lunar gravity is if you watch films of astronauts moonwalking.
This was an enormous thing Newton did - to invent a new kind of math to build a model that described in the same formula the observed motion of both falling objects on Earth and the planets in the heavens.
BUT unfortunately, Newtonian gravity falls apart when we try to combine it with what we've just learned about Special Relativity.