Question: I have the following two matrices:

1) $$\begin{bmatrix}1&0&0\\ 0&1&1\\ 0&0&0\\0&0&0 \end{bmatrix}$$

I believe this matrix is in the form of reduced row echelon form but I'm a bit unsure.

2) $$\begin{bmatrix}1&2&6&0&0\\0&0&1&2&-2\\0&0&0&0&1\end{bmatrix}$$

I believe this matrix is in row echelon form.

I'm still not 100% sure what defines a row echelon form and reduced row echelon form. I've tried looking everywhere for a clear understanding of it, but I just could grasp it 100%.

Answer: 
That second matrix is
$$
\begin{bmatrix}
\color{blue}1&2&\color{red}6 &0& 0\\
0 &0&\color{blue}1&2&\color{red}{-2}\\
0 &0& 0&0&\color{blue} 1\end{bmatrix}
$$
The blue $1$s are **pivots**, that is, the leading $1$ of the row. In order for the matrix to be in reduced row-echelon form (rref), the numbers above and below the pivots must be zero. Because the red numbers are non-zero, this matrix is not in rref.
The first matrix is already in reduced row echelon form. The second matrix can be simplified to

$$\begin{bmatrix}
1 & 2 & 0 & -12 & 0\\
0 & 0 & 1 & 2 & 0\\
0 & 0 & 0 & 0 & 1
\end{bmatrix}$$