Johannes Kepler (1571-1630)
The most radical thing Kepler did was to remove the assumption of circular motions, a motion viewed by the Greeks and Copernicus as sacred. This simple generalization allowed Kepler to make a model which was consistent the the data acquired by Tycho.

Kepler's Laws of Planetary Motion
Law I: Law of Ellipses

Each planet moves about the Sun in an orbit that is an ellipse with the Sun at one focus of the ellipse. The eccentricity is defined as the ratio of the distance of the focus (from the center of the ellipse) to the semi-major axis (c/a in the notation of the figure). For a circle, the eccentricity is then 0. For the planets in our Solar System, the eccentricities range from 0.007 (Venus) to 0.248 (Pluto).
The orbits are nearly circular, however. The distortions are quite small and visually hard to see, in general:

Law II: Law of Equal Areas

A straight line joining a planet and the Sun sweeps out equal areas in equal amounts of time. The consequence of this is that a planet moves fastest when it is closest to the Sun.

Law III: Harmonic Law

The square of the sidereal revolution period of a planet is proportional to the cube of the size of its orbit (actually, semi-major axis of its orbit). The planets closer to the Sun have the shortest sidereal orbital periods. For the Solar System, P² (years) = a³ (Astronomical Units).

The above is a log-log plot, log P vs. log a

In tabular form,

Note that the last two entries in the above table are discrepant.

Application
The asteroid belt sits between the orbits of Mars and Jupiter. Compare the orbital period for an object that has orbit of size 4 A.U. to the orbital period of the Earth. Using Kepler's 3rd Law we have that the square of the orbital period measured in years is
P² = a³ = 4³ = 64
So the orbital period of the asteroid measured in years is the square root of 64, that is, the orbital period of the asteroid is 8 years.

Comments:

Comparison to data and Copernicus
Titius-Bode Law
Galilean Moons of Jupiter