Suppose we have two events $(x_1,y_1,z_1,t_1)$ and $(x_2,y_2,z_2,t_2)$, then we can define
$$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$
which is called the spacetime interval. The first event occurs at the point with coordinates $(x_1,y_1,z_1)$ and the second at the point with coordinates $(x_2,y_2,z_2)$ which implies that the quantity
$$r^2 = \Delta x^2+\Delta y^2+\Delta z^2$$
is the square of the separation between the points where the events occur. In that case the spacetime interval becomes $\Delta s^2 = r^2 - c^2\Delta t^2$. The first event occurs at time $t_1$ and the second at time $t_2$ so that $c\Delta t$ is the distance light travels on that interval of time.
In that case, $\Delta s^2$ seems to be comparing the distance light travels between the occurance of the events with their spatial separation. The definition done is then the following:
If $\Delta s^2 <0$ then $r^2 < c^2\Delta t^2$ and the spatial separation is less than the distance light travels and the interval is called timelike.
If $\Delta s^2 = 0$ then $r^2 = c^2\Delta t^2$ and the spatial separation is equal to the distance light travels and the interval is called lightlike.
If $\Delta s^2 >0$ then $r^2 > c^2\Delta t^2$ and the spatial separation is greater than the distance light travels and the interval is called spacelike.
These are just mathematical definitions. What, however, is the physical intuition behind? I mean, what an interval being timelike, lightlike or spacelike means?