In a book that I'm currently reading it is stated that
If $\det (R) = +1$ (where $R$ is an orthogonal matrix) then the matrix $R$ represents a pure rotation [...] If $\det(R) = -1$ then the matrix $R$ represents a pure rotation + reflection.
I am very confused by this description; I still do not know what their difference really is (is it that when $\det (R) = 1$ we are only dealing with rotations and not reflections (but aren't reflections some form of rotation?); what does the $+$ sign mean (does it mean "or" or "a reflection followed by a rotation")). I would appreciate it if someone would explicitly describe their differences for me.
The answer to this question is no, and I suspect that's where your confusion lies. If you take an asymmetric figure such as the figure ${\large\mathrm L}$ and rotated it, you would never be able to get it to look like its horizontal reflection (try it!). So if $\det R=1$, then $R$ corresponds to a rotation of the plane, whereas if $\det R=-1$, then it corresponds to either just a reflection, or a reflection followed by a rotation of the plane (in the $2\mathrm{D}$ case).