The group of the isometries of the square (the dihedral group $D_8$) is generated by the rotation $\rho$ and the reflection $\sigma$.
Now I have no problem with understanding the rotation but the reflections are more ambiguous. In a square, a reflection is made according to an axis. In the isometries of the square, there exists 4 different reflections. If 1,2,3,4 are the corners of the square (clockwise) then a reflection can be made according to
the axis that goes through 1 and 3
the axis that goes through 2 and 4
the axis that goes through the center of [1,2] and the center of [3,4]
the axis that goes through the center of [1, 4] and [2, 3]
How do we know according to which axis the reflections are made
You need to define $\sigma$ to be a particular one of the four reflections—it doesn't matter which. This is really no different from having to define the rotation. (It can be defined either as a $90^\circ$ clockwise rotation or as a $90^\circ$ counterclockwise rotation.) No matter what definitions you adopt, it will be the case that elements of the form $\rho^j$ are rotations and elements of the form $\rho^j\sigma$ are reflections.
For example, if $\sigma$ is defined to be the reflection about the axis through the top-left and bottom-right corners of the square, and $\rho$ is defined to be the $90^\circ$ clockwise rotation, then $\rho\sigma$, which I will interpret as $\sigma$ followed by $\rho$, is the reflection about the vertical axis of the square. Similarly, $\rho^2\sigma$ is the reflection about the axis through the top-right and bottom-left corners of the square, and $\rho^3\sigma$ is the reflection about the horizontal axis of the square.