Suppose $A$ is an $m\times n$ matrix. Extend $A$ on its right side with the addition of an $m\times m$ identity matrix to form an $m\times (n + m)$ matrix $M$. Use row operations to bring $M$ to reduced row-echelon form and call the result $N$. $N$ is the extended reduced row-echelon form of $A$, and we will standardize on names for five submatrices ($B$, $C$, $J$, $K$, $L$) of $N$.
Let $B$ denote the $m\times n$ matrix formed from the first $n$ columns of $N$ and let $J$ denote the $m\times m$ matrix formed from the last $m$ columns of $N$. Suppose that $B$ has $r$ nonzero rows. Further partition $N$ by letting $C$ denote the $r\times n$ matrix formed from all of the nonzero rows of $B$. Let $K$ be the $r\times m$ matrix formed from the first $r$ rows of $J$, while $L$ will be the $(m-r)\times m$ matrix formed from the bottom $m-r$ rows of $J$. Pictorially, \begin{equation*} M=[A\vert I_m] \rref N=[B\vert J] = \left[\begin{array}{c|c}C&K\\\hline0&L\end{array}\right] \end{equation*}