Why is correlation formula the way it is? Or Say how it formed? I am really being confused by why the correlation formula is called the correlation of two variables $X$ and $Y$. Also how is it derived? 
The part where we divide covariance by product of standard derivation of $X$ and standard derivation of $Y$ is the most confusing for me. Please explain the reasons or provide some good source for such things. 
 A: Maybe going back to the notion of covariance would help.
Say we have two random variables $X$ and $Y$, with a certain number $n$ of independent realizations $x_1,x_2,\dots x_n$ and $y_1,y_2,\dots y_n$. We know that the formula for the sample covariance is
$$\sigma_{xy} =\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})$$
where $\bar{x}$ and $\bar{y}$ are respectively sample means for $X$ and $Y$.
Now, thanks to the Cauchy-Schwarz inequality, we have that the sample covariance is bounded by the product of the standard deviations of the two random variables, which I will denote with $\sigma_x$ and $\sigma_y$. We have then that
$$-\sigma_x\sigma_y \leq \sigma_{xy}\leq \sigma_x\sigma_y$$
Now divide all terms in the inequality by $\sigma_x\sigma_y$ (which are, by construction, non-negative) and you have the formula for correlation
$$-\frac{\sigma_x\sigma_y}{\sigma_x\sigma_y} \leq \frac{\sigma_{xy}}{\sigma_x\sigma_y}\leq \frac{\sigma_x\sigma_y}{\sigma_x\sigma_y} $$
$$-1 \leq \frac{\sigma_{xy}}{\sigma_x\sigma_y} \leq 1$$
with the correct bounds, $-1$ and $1$. If you grasp the notion of covariance, then you'll surely see that it is simply a standardized version of the latter.
