Why do planets move on ellipses?

Johannes Kepler found the ellipses in 1609, by fitting Tycho Brahe's careful observations of Mars. He did not know why. Isaac Newton explained it in the Principia (1687): if the force pulling a planet towards the Sun has strength proportional to $1/r^2$, then the planet's path is a conic section with the Sun at a focus. In polar coordinates centred on the Sun,

$$r = \frac{\ell}{1 + e\cos\theta},$$

where $e$ is the eccentricity: $e = 0$ gives a circle, $0 < e < 1$ an ellipse, $e = 1$ a parabola and $e > 1$ a hyperbola. Which one occurs depends on the planet's energy. Comets on long ellipses return; an object with too much energy passes once along a hyperbola and leaves — as the interstellar visitor 'Oumuamua did in 2017.

Ellipses are special. In 1873 Joseph Bertrand proved that among forces that depend only on distance, only two make every bounded orbit close up: the inverse-square law and a spring-like force proportional to $r$. With any other law, orbits trace rosettes.

Real orbits are not perfect ellipses: the other planets tug, and the ellipse of Mercury slowly turns. Almost all of that turning is explained by Newton's law, but an extra $43$ arcseconds per century was explained only in 1915, by Einstein's general relativity.