Short answers, pulled from the story.
Kepler's laws of planetary motion are three laws describing how planets orbit the Sun. The first states that a planet's orbit is an ellipse with the Sun at one focus. The second states that a line joining a planet and the Sun sweeps out equal areas in equal times. The third states that the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit.
Johannes Kepler discovered the laws, publishing them between 1608 and 1621 in three works: Astronomia nova, Harmonice Mundi, and Epitome Astronomiae Copernicanae. He based them on the precise astronomical observations of Tycho Brahe.
Kepler published his first law in Astronomia nova in 1609 and his third law in Harmonice Mundi in 1619. He presented the second law in its modern form in his Epitome Astronomiae Copernicanae of 1621.
Kepler discovered elliptical orbits by analyzing Tycho Brahe's highly precise observations of Mars. He could not fit a circle to Mars' orbit, which has the highest eccentricity of any planet except Mercury, and concluded the orbit was an ellipse.
Copernicus held that planetary orbits were circles with epicycles, that the Sun sat approximately at the centre, and that planetary speed was constant. Kepler corrected this, showing orbits are ellipses with the Sun at a focal point and that only the area speed of a planet stays constant.
Isaac Newton showed in 1687 that Kepler's laws follow from his law of universal gravitation, with the acceleration of a planet directed toward the Sun and falling off as the inverse square of distance. Because planet masses are small compared with the Sun, orbits conform approximately to Kepler's laws, which became the basis for computing planetary positions over time.