Gravity as a curved path
Gravity is not a force that "reaches out" and grabs objects — Einstein showed it is a curvature of spacetime caused by mass. Objects follow the straightest possible path through curved spacetime — and around a massive body like the Sun, those straight paths in curved spacetime are ellipses. Newton's earlier description — that gravity is a force pulling objects toward each other — gives the same orbits and is accurate enough for most engineering purposes.
The critical quantity is orbital speed. At a given altitude above a planet, there is exactly one circular orbital speed. Below that speed, the object spirals inward; above it, the object rises on an ellipse. At √2 × circular speed (escape velocity), the object leaves on a parabola and never returns. This is why rocket staging matters — each stage adds the speed needed to reach the next orbit.
Kepler's three laws, briefly
These laws apply to any two-body gravitational system — moons orbiting planets, binary stars, exoplanets, and artificial satellites. See how they're used in exoplanet detection, where the radial velocity method detects the orbital wobble they cause in a star.