I was reading about star polygon graphs from the following link:
http://mathworld.wolfram.com/StarPolygon.html.
As far as I noticed I felt that whenever $d$ is a proper divisor of $n$, then we get $d$ copies of cycle of length $n/d$ where $~$ $d<\lfloor n\rfloor$. Is my observation correct? Kindly rectify me if I am wrong somewhere.
Yes, your interpretation is consistent with the MathWorld entry.
However, it should be noted that not everyone uses this interpretation. In particular, Grünbaum and others (like me) take the model regular $\{n/d\}$-gon to have its $k$-th vertex (starting at the $0$-th) at coordinates $$\left(\;\cos \frac{2\pi dk}{n}\;,\; \sin\frac{2\pi dk}{n} \;\right)$$ With this view, a $\{12/4\}$-gon, for instance, isn't a compound of four separate triangles in the MathWorld sense; it's a dodecagon that wraps around a single triangular cycle four times. See some related thoughts in this answer to the question "What is a Hexagon?".