Answer
It would be more proper to ask, "what is the mass of the earth"^*. The quick answer is: approximately 6,000,000,000,000,000,000,000,000 (6E+24) kilograms.

The interesting sub-question is, "how did anyone figure that out?" It's not like the planet steps onto the scale each morning before it takes a shower! The measurement of the planet's weight is derived from the gravitational attraction that the earth has for objects near it.

It turns out that any two masses have a gravitational attraction for one another. If you put 2 bowling balls near each other, they will attract one another gravitationally. The attraction is extremely slight, but if your instruments are sensitive enough you can measure the gravitational attraction that two bowling balls have on one another. From that measurment you could determine the mass of the two objects. The same is true for two golf balls, but the attraction is even slighter because the amount of gravitational force depends on the amount of mass in the objects.

Newton showed that, for spherical objects, you can make the simplifying assumption that all of the object's mass is concentrated at the center of the sphere. The following equation expresses the gravitational attraction that 2 spherical objects have on one another:

F = G * M1 * M2 / R²

Assume that earth is one of the masses (M1) and a 1 kg sphere is the other (M2). The force between them is 9.8 kg*m/s² - we can calculate this force by dropping the 1 kg sphere and measuring the acceleration that the earth's gravitational field applies to it (9.8 m/s²). The radius of the earth is 6,400,000 meters. If you plug all of these values in and solve for M1, you find that the mass of the earth is 6,000,000,000,000,000,000,000,000 (6E+24) kilograms.

[* - It is "more proper" to ask about mass rather than weight because weight is a force that requires a gravitational field to determine. You can take a bowling ball and weigh it on the earth and on the moon. The weight on the moon will be 1/6 that on the earth, but the amount of mass is the same in both places. To weigh the earth we would need to know in what object's gravitational field we want to calculate the weight. The mass of the earth, on the other hand, is a constant.]

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