What does $\ dx^2$ mean? While writing the second derivative of y, $\frac{d^2y}{dx^2}$ what does the symbol $dx^2$ signify? I know that in case of the first derivative $dy$ means change in y and $dx$ means change in y and also that $d^2$ means change of change of y.

It does not have a meaning on its own. $\dfrac{d^2 y}{dx^2}$ is just an abbreviation for $\dfrac{d \left(dy/dx\right)}{dx}$, that is the derivative of $\dfrac{dy}{dx}$.