Elliptical orbits and equal areas
Kepler's first law states that planetary orbits are ellipses with the Sun at one focus, not circles. The eccentricity (how elongated the ellipse is) varies: Mercury's orbit is highly eccentric, Earth's nearly circular. An ellipse has a closest point (perihelion) and farthest point (aphelion); the difference between them determines orbital eccentricity.
Kepler's second law is the equal-areas principle: a line from the Sun to a planet sweeps equal areas in equal times. This means planets move faster near perihelion (closest to the Sun) and slower near aphelion (farthest). Earth moves fastest in early January (near perihelion) and slowest in early July (near aphelion). The law follows from conservation of angular momentum and requires no external forces.
The orbital period formula
Kepler's third law relates orbital period to orbital size: the square of the period is proportional to the cube of the semi-major axis (the ellipse's characteristic size). Mathematically, T^2 proportional to a^3. This allows measurement of orbital distances from observed periods: if a planet's period is known and the Earth's period and distance are known, the planet's distance follows without direct observation.
The third law applies to any orbit: planets around the Sun, moons around planets, and asteroids around the Sun all obey it. It was later explained by Newton's law of gravitation and applies universally to binary stars, exoplanet systems, and black hole accretion disks.