How heavy is the Earth? What method is used to figure this out?
The mass of the earth is 6 x 1024 kilograms.
The interesting sub-question is, "how did anyone figured that out?"
The answer lies in Newton's universal law of gravity. According to Newton's law, every two massive objects attract each other at a force that is directly proportional to their mass, and inversely proportional to the square of the distance between their centers. In formula, it would read: , where G is a universal constant of nature, known (surprise, surprise) as the universal gravitational constant.
Furthermore, according to Newton's second law of motion, the force causes a body to accelerate (change its velocity), with the proportionality constant equals to the object's mass. In formula form, it reads F = m a.
Combined together, if we take an object and let it fall freely subject only to the earth's gravity, it will accelerate in accordance to
Thus, by measuring the acceleration, we obtain a direct measurement of the earth's mass divided by its radius (squared) and multiplied by the gravitational constant. Note that this formula further implies that the acceleration is independent on the mass of the object- heavy and light masses will accelerate at the same rate! This was demonstrated by the famous experiment made by Galileo, throwing two objects, one light and one heavy from the leaning tower of Pisa proving that they arrive at the ground at the same time.
So in order to determine the earth's mass, it is left to determine (1) the earth's radius, and (2) the universal gravitational constant.
Measurements of the earth's radius dates back a long time ago - to about 240BC. Such a measurement was first conducted by the Greek astronomer Eratosthenes. Eratosthenes knew that at local noon on the summer solstice (the longest day of the year) in Syene (today, it's the modern city of Aswan, Egypt), the Sun was directly overhead. He knew this because the shadow of someone looking down a deep well at that time in Syene blocked the reflection of the Sun on the water. He measured the Sun's angle of elevation at noon on the same day in Alexandria. The method of measurement was to make a scale drawing of that triangle which included a right angle between a vertical rod and its shadow. This turned out to be 1/50th of a full angular revolution. Taking the Earth as spherical, and knowing both the distance and direction of Syene, he concluded that the Earth's circumference was fifty times that distance. Despite his very primitive method (e.g., at that time there was no accurate way of measuring the distance between Alexandria and Syene), Eratosthenes result was only about 10% different than the true result.
Thus, the earth's radius - about 6400 km, was pretty well known at Newton times.
As for measuring the gravitational constant, that had to wait until Henri Cavendish measured it in his famous experiment in 1798. His apparatus contained a torsion balance made of a six-foot (1.8 m) wooden rod suspended from a wire, with a 2-inch (51 mm) diameter 1.61-pound (0.73 kg) lead sphere attached to each end. Two 12-inch (300 mm) 348-pound (158 kg) lead balls were located near the smaller balls, about 9 inches (230 mm) away, and held in place with a separate suspension system. The experiment measured the faint gravitational attraction between the small balls and the larger ones.
The two large balls were positioned on alternate sides of the horizontal wooden arm of the balance. Their mutual attraction to the small balls caused the arm to rotate, twisting the wire supporting the arm. The arm stopped rotating when it reached an angle where the twisting force of the wire balanced the combined gravitational force of attraction between the large and small lead spheres. By measuring the angle of the rod and knowing the twisting force (torque per unit radian) of the wire for a given angle, Cavendish was able to determine the force between the pairs of masses. Since the gravitational force of the Earth on the small ball could be measured directly by weighing it, the ratio of the two forces allowed the density of the earth to be calculated, using Newton's law of gravitation.
Cavendish's equipment was remarkably sensitive for its time. The motion of the rod was only about 0.16 inches (4.1 mm). Cavendish was able to measure this small deflection to an accuracy of better than one hundredth of an inch using a special device known as vernier scales on the ends of the rod. Cavendish's result was extremely accurate - the measured value of the gravitational constant, hence of the earth's mass was only about 1% different than the modern value!.