When writing the system $\tag{1} \eqalign{ \color{maroon}{-1}\cdot c_1+\color{darkgreen}4c_2&=\color{darkblue}{-12}\cr \color{maroon}2c_1\color{darkgreen}{-6}c_2&=\color{darkblue}{20} } $ as an augmented matrix: $\tag{2} \left[ \matrix{\color{maroon}{ -1}&\color{darkgreen}4 \cr\color{maroon} 2&\color{darkgreen}{-6}} \biggl| \matrix{\color{darkblue}{-12}\cr \color{darkblue}{20}}\right] $
You're essentially just writing the original system down, but you are not writing the variables or the equality sign. In $(2)$, the red terms are the coefficients of the variable $c_1$ in the system of equations $(1)$. So column one of $(2)$ is the ''$c_1$-column". Similarly, column two of $(2)$ is the ''$c_2$-column" and the third column corresponds to the constants on the right hand side of the equalities of the system of equations $(1)$.
Performing a row operation on the matrix $(2)$ corresponds to performing the same operation on the system.
A reduced form of the augmented matrix $(2)$ is
$\tag{3} \left[ \matrix{\color{maroon}{ -1}&\color{darkgreen}4 \cr\color{maroon} 0&\color{darkgreen}{2}} \biggl| \matrix{\color{darkblue}{-12}\cr \color{darkblue}{-4}}\right] $ This was obtained by adding twice row one to row two. (Note that this corresponds to adding twice equation one to equation two in the system of equations $(1)$.)
Translating the matrix $(3)$ back to the corresponding system of equations:
Row one of $(3)$ gives $\color{maroon}{-1}c_1+\color{darkgreen}4c_2=\color{darkblue}{-12}$
Row two of $(3)$ gives $\color{maroon} 0c_1+\color{darkgreen}2c_2=\color{darkblue}{-4}.$
You could reduce $(3)$ further to
$\tag{4} \left[ \matrix{\color{maroon}{ 1}&\color{darkgreen}0 \cr\color{maroon} 0&\color{darkgreen}{1}} \biggl| \matrix{\color{darkblue}{ 4}\cr \color{darkblue}{-2}}\right] $ And the corresponding system would be $\eqalign{\color{maroon}1c_1 +\color{darkgreen}0 c_2&=\color{darkblue}4\cr \color{maroon}0c_1+\color{darkgreen}1c_2&=\color{darkblue}{-2} }$, or simply $c_1=4$, $c_2=-2$.
Just keep in the back of your mind that writing
$ \left[ \matrix{\color{maroon}{ a}&\color{darkgreen}b \cr\color{maroon} c&\color{darkgreen}{d}} \biggl| \matrix{\color{darkblue}{ e}\cr \color{darkblue}{f}}\right] $
is shorthand notation for writing $ \eqalign{\color{maroon}a\,c_1 +\color{darkgreen}b\, c_2&=\color{darkblue}e\cr \color{maroon}c\,c_1+\color{darkgreen}d\,c_2&=\color{darkblue}{f} }. $