Weight
In modern scientific usage, weight and mass are fundamentally different quantities: mass is an intrinsic property of matter, whereas weight is a "force" that results from the action of gravity on matter: it measures how strongly the force of gravity pulls on that matter. However, in most practical everyday situations the word "weight" is used when, strictly, "mass" is meant. For example, most people would say that an object "weighs one kilogram", even though the kilogram is a unit of mass.
Earth mass
The mass of Earth is measured indirectly by determining other quantities such as Earth's density, gravity, or gravitational constant.
The first measurement in the 1770s Schiehallion experiment resulted in a value about 20% too low.
The Cavendish experiment of 1798 found the correct value within 1%.
Uncertainty was reduced to about 0.2% by the 1890s, 
to 0.1% by 1930.