It would be more proper to ask, "What is the mass of planet Earth?" The quick answer to that is approximately 6,000,000, 000,000,000,000,000,000 (6 x 1024) 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 two 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 measurement, you could determine the mass of the two objects. The same is true for two planets in space.
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 two spherical objects have on one another:
F = G(M1*M^2/R2)