How is the mass of the Earth determined? According to textbook knowledge, the mass of the Earth is about $6 × 10^{24}\,\mathrm{kg}$.  How is this number determined when one cannot just weight the Earth using regular scales?
According to Newton's Law of Gravity based on attractive force (gravitational force) that two masses exert on each other:
$$F=\frac{GmM}{r^2}$$
Where:

$F$ is the gravitational force
$G = 6.67 \times 10^{-11}\ \mathrm{m}^3\ \mathrm{kg}^{-1}\ \mathrm{s}^{-2}$ is a constant of proportionality
$M$ and $m$ are the two masses exerting the forces
$r$ is the distance between the two centers of mass.

From Newton's second law of motion:
$$F=ma$$
Where:

$F$ is the force applied to an object
$m$ is the mass of the object
$a$ is its acceleration due to the force.

Equating both the equations:
$$F = \frac{GmM}{r^2} = ma$$
$$\frac{GM}{r^2}= a$$ (The $m$'s canceled out.)
Now solve for $M$, the mass of the Earth.
$$M = \frac{ar^2}{G}$$
Where $a = 9.8\ \mathrm{m}\ \mathrm{s}^{-2}$, $r = 6.4 \times 10^6\ \mathrm{m}$, and $G = 6.67 \times 10^{-11}\ \mathrm{m}^3\ \mathrm{kg}^{-1}\ \mathrm{s}^{-2}$.
$$M = 9.8 \times (6.4 \times 10^6)^2/(6.67 \times 10^{-11})\ \mathrm{kg}$$

Hence,
$M = 6.0 \times 10^{24}\ \mathrm{kg}$