Why is $i^2$ equal to $-1$? In this KhanAcademy link at 2:25, Sal (the narrator) says that $i^2$ is negative 1 and he didn't explain why. 

Why is this so? What is the intuition behind it?
It is defined that way.

I believe the proper definition of $i$ is $i^2 = -1$ and not $i = \sqrt{-1}$ as is commonly stated. 

It is useful in many deep areas of mathematics, but as a starting point, it serves as a solution to $x^2 + 1 = 0$. 

I recommend looking deeper into the theory of complex numbers, particularly the polar form.

To help you understand further. Realize that we have many sets of numbers such as the natural numbers, the rationals, and the reals. The complex number system is just another set and each complex number can be defined as an ordered pair $a + ib$ or $(a,b)$ where $a,b \in \mathbb{R}$.
A note:
Since $i^2 = -1$,
there is no way to tell the difference
between
$i$ and $-i$.