Why do things here on Earth fall down? I want to have an answer with that question above for my physics lesson. I really don't have an idea about it, so, I ask help from you guys and hope that someone can help me with it.
 A: In Newtonian mechanics: 
Things fall due to the gravitational force of Earth. This force between 2 masses $M$ and $M'$ is given by: $$F = G\frac{M \cdot M'}{r^2}$$.
This force is a central conservative force due to mutual attraction.
The rise of General Relativity:
Albert Einstein,in 1915, looked at the gravity through new eyes. For him, the fact that all objects fall toward the earth with $g$,whatever their size,implied that this must be in some truly profound way a geometric and kinematic result,not a dynamic one. He regarded it as being on a par with Galileo's law of inertia,which was accurate in straight-line motion.
Building on these ideas,Einstein developed the theory that 

Objects which are at rest tend to remain at rest, and objects which are moving tend to move along geodesic paths - that is to say,the most economical way of getting from one point to other, with uniform motion,unless some force acts on them.

Objects fall simply because massive objects like Earth modifies the geometry locally so that the shortest straight lines become geodesics. The state of affairs in the vicinity of a massive object is,in this view,to be interpreted not in terms of a gravitational field of a force but in terms of a  CURVATURE OF SPACE.
A: In 1600 Newton discovered a formula that could explain why things fall down on earth
things fall due to the gravitational force of the Earth. The formula for the intensity of the force is:  $$F = G \dfrac{m_1 \cdot m_2}{r^2}$$
where $m_1$ and $m_2$ are the 2 masses of the object, $r$ is the distance between them and G is a constant called Universal Gravitational Constant (also discovered by Newton) and is $6,673 × 10^{−11}$
This is the GENERAL formula. 
Now consider the gravitational force of the Earth; if we plug in the equation the mass of the Earth and the distance of an object on the surface of the Earth and the centre of it (which is just the radius of the Earth) we get:
$$F=G \frac {M_{earth} \cdot m_{object}}{r_{earth}^2}=6,673 × 10^{−11} \cdot \frac{(5,9736 \cdot 10^{36})\cdot m}{(6372,795 \cdot 10^3)^2}\approx m\cdot 9,81 $$
$9,81$ is the result of all the calculation and physicist call this $g$ which simplifies all the calculations. If you want to calculate the gravity force on an object by the Earth you just have to take the mass of the object and multiply it by $9.81$
