On the notation for the Jacobian using indices A contravariant vector is an object that is usually written with a superscript and it is defined by the "transformation law":
$$V^{'i} = \frac{\partial x^{' i}}{\partial x^j} V^j $$
where $i,j = 0,1,2,3$.
In the definition above the fractional terms is referred to as the Jacobian. Now my confusion is the following: I understand that in the definition above really represent 4 different equations (due to the Einstein summation convention), but why does fractional term represent a matrix (the Jacobian).
 A: You've probably seen other Jacobians in some class on multivariate calculus, when computing an integral and changing variables. The expression $\frac{\partial x'^i}{\partial x^j}$ has two indices, so we can regard it as a matrix $J^i_j$ that connects the primed and unprimed coordinate systems. If we write a position vector in these coordinates as
$$
[x^\mu]=\left(\begin{array}{c}
x_1\\x_2\\ \vdots\\x_n
\end{array}\right), \ \ \ \ 
[x'^\nu]=\left(\begin{array}{c}
x'_1\\x'_2\\ \vdots\\x'_n
\end{array}\right),
$$
where $\mu,\nu=1,\cdots,n$, 
the transformation from  one to another is a matrix
$$
\left[\frac{\partial x'^\nu}{\partial x^\mu}\right]=\left(\begin{array}{cccc}
\frac{\partial x'^1}{\partial x^1} & \frac{\partial x'^1}{\partial x^2} & \cdots & \frac{\partial x'^1}{\partial x^n}\\\frac{\partial x'^2}{\partial x^1} & \ddots\\ \vdots & & \ddots\\ \frac{\partial x'^n}{\partial x^1} & & &\frac{\partial x'^n}{\partial x^n}
\end{array}\right),
$$
so that
$$
x'^\nu = \frac{\partial x'^\nu}{\partial x^\mu}x^\mu
$$ 
in the sense of matrix multiplication (as you said, $n$ equations).
