If you have a 3x3 matrix representing some transformation, you will actually have the X,Y,Z vectors of that transformation in the rows or columns (depending on if it's a row major or column major matrix).
In other words, if you have a 3x3, you can look at it and immediately get the up, right(*), forward vectors (asterisk due to handedness, it could be the left vector).
3x3 Details:
let's say you have a 3x3 matrix $M$:
$
M_{ij} =
\begin{bmatrix}
M_{1,1} & M_{1,2} & M_{1,3} \\
M_{2,1} & M_{2,2} & M_{2,3} \\
M_{3,1} & M_{3,2} & M_{3,3} \\
\end{bmatrix}
$
Let's see what happens when you multiply that by a vector $\begin{bmatrix}1,0,0 \end{bmatrix}$:
$
\begin{bmatrix}
M_{1,1} & M_{1,2} & M_{1,3} \\
M_{2,1} & M_{2,2} & M_{2,3} \\
M_{3,1} & M_{3,2} & M_{3,3} \\
\end{bmatrix}
*
\begin{bmatrix}
1 \\
0 \\
0 \\
\end{bmatrix}
=
\begin{bmatrix}
M_{1,1} \\
M_{2,1} \\
M_{3,1} \\
\end{bmatrix}
$
Now let's try vector $\begin{bmatrix}0,1,0\end{bmatrix}$:
$
\begin{bmatrix}
M_{1,1} & M_{1,2} & M_{1,3} \\
M_{2,1} & M_{2,2} & M_{2,3} \\
M_{3,1} & M_{3,2} & M_{3,3} \\
\end{bmatrix}
*
\begin{bmatrix}
0 \\
1 \\
0 \\
\end{bmatrix}
=
\begin{bmatrix}
M_{1,2} \\
M_{2,2} \\
M_{3,2} \\
\end{bmatrix}
$
And lastly, vector $\begin{bmatrix}0,0,1\end{bmatrix}$:
$
\begin{bmatrix}
M_{1,1} & M_{1,2} & M_{1,3} \\
M_{2,1} & M_{2,2} & M_{2,3} \\
M_{3,1} & M_{3,2} & M_{3,3} \\
\end{bmatrix}
*
\begin{bmatrix}
0 \\
0 \\
1 \\
\end{bmatrix}
=
\begin{bmatrix}
M_{1,3} \\
M_{2,3} \\
M_{3,3} \\
\end{bmatrix}
$
This shows that whatever matrix $M$ is, the first column is the X axis vector of the coordinate space defined by the matrix. The second column is the Y axis vector, and the third column is the Z axis vector.
These vectors are in global space.
$
M =
\begin{bmatrix}
X_x & Y_x & Z_x \\
X_y & Y_y & Z_y \\
X_z & Y_z & Z_z \\
\end{bmatrix}
$
You can even see that this is true in an identity matrix, where the X,Y and Z vectors are just what you'd expect them to be:
$
M =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
$
A 3x3 matrix is much like a vector in that it describes directions but has no position.
If you have a 4x4 matrix that represents a 3d transform and uses homogeneous coordinates for a translation, you can then use that to get the X,Y,Z orientation vectors, but also can get the translation from the origin.
Hopefully this is helpful, and not misleading for less common matrix types. I saw an answer deleted which said similar, so fingers crossed :P
