One rule for every scale

The law. Newton's law of gravitation says that any two masses attract each other with a force F = G·m₁·m₂ / r². G is the gravitational constant, 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻² [1]. The force grows with each mass and falls with the square of the distance between them. Double the distance and the force becomes a quarter; triple the distance and it becomes a ninth.

Equal and opposite. The Sun pulls the Earth with a force of about 3.5 × 10²² newtons, and the Earth pulls the Sun with exactly the same force, as Newton's third law requires. The reason the Earth moves so much more is that its mass is small: a = F / m, and the Sun's mass is about 333,000 times the Earth's [2].

Mass and weight. Mass is how much matter a body has; weight is the force with which gravity pulls on it. Your mass is the same everywhere; your weight on the Moon, whose surface gravity is about a sixth of the Earth's, would be about a sixth [3].

Building a solar system. Gravity is not only a force between finished bodies. The early universe was almost smooth, to about one part in 100,000 [4]. Wherever the gas was slightly denser it pulled in more gas. About 4.6 billion years ago a cloud of gas and dust collapsed under its own gravity into a spinning disk with the Sun at its centre; dust grains stuck together and their growing gravity swept up more material, building planets [5].

Sources

1. CODATA recommended value: Newtonian constant of gravitation (NIST), https://physics.nist.gov/cgi-bin/cuu/Value?bg
2. Sun Fact Sheet (NASA GSFC NSSDC), https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
3. Moon Fact Sheet (NASA GSFC NSSDC), https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html
4. COBE science (NASA Science), https://science.nasa.gov/mission/cobe/science/
5. Sun: Facts (NASA Science), https://science.nasa.gov/sun/facts/