The two previous writeups contain some slight errors.

First Law: The orbit of a planet about the sun is an ellipse with the Sun at one focus.

Second Law: A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

Third Law: The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit.

In the case of our solar system, the Third Law can be stated mathematically as:

P2 = a3

P = sidereal period of planet (in years)
a = planet's semimajor axis (in AU)

Newton's formulation of the Third Law is more complicated but allows calculations for any pair of bodies, not just the Sun and the planets. It is stated as:

P2 = (4pi2a3)/(G(m1 + m2))

P = sidereal period of orbit (in seconds)
a = semimajor axis of orbit (in meters)
m1, m2 = masses of two bodies (in kg)
G = gravitational constant (approximately 6.67 x 10-11 m3kg-1s-2

This can be used to calculate the period of any pair of orbiting bodies. However, this is usually easily determined by empirical observation. A more useful formulation is:

m1 + m2 = (4pi2a3)/(GP2)

This allows the combined mass of any pair of bodies to be calculated from their orbital distance and sidereal period. As the ratio of their masses can be calculated from the position of their centre of gravity, the individual masses can hence be determined.