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A matrix is in row echelon form, or REF, if it has all of the following three properties:

The zero rows of the matrix (if any) are all at the bottom of the matrix.
In every non-zero row of the matrix, the leading entry is $1$.
If row $i$ and row $(i+1)$ are both non-zero, then the leading entry in row $(i+1)$ is to the right of the leading entry in row $i$. <html><br /></html>In other words, as you go down the rows, the leading entries must go to the right.

For example, $\left[\begin{smallmatrix} 1&2&3&4&5\\0&1&2&3&4\\0&0&1&2&3\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 1&2&3&4&5\\0&1&2&3&4\\0&0&1&2&3\\0&0&0&0&0\end{smallmatrix}\right]$ are both in REF, but

$\left[\begin{smallmatrix} 1&2&3&4&5\\0&0&0&0&0\\0&0&1&2&3\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 1&2&3&4&5\\0&0&0&0&0\\0&0&1&2&3\\0&0&0&0&0\end{smallmatrix}\right]$ are not in REF, since they each have a zero row which isn't at the bottom;
$\left[\begin{smallmatrix} 1&2&3&4&5\\0&2&3&4&1\\0&0&1&2&3\end{smallmatrix}\right]$ is not in REF, since the leading entry on the second row isn't $1$;
$\left[\begin{smallmatrix} 0&1&2&3&4\\1&2&3&4&5\\0&0&1&2&3\end{smallmatrix}\right]$ is not in REF, since the leading entry in row $2$ is not to the right of the leading entry in row $1$.