Special Relativity is based on the invariance of a quantity called the proper time, $\tau$, which is the time measured by a freely moving (i.e. not accelerated) observer. The proper time is defined by:
$$ c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 $$
This is similar to Pythagoras' theorem as learned by generations of schoolchildren, except that it includes time (converted to a distance by multiplying by $c$) and it has a mixture of plus and minus signs. The mixture of signs is responsible for all the weird effects like time dilation and length contraction, and because there is a mixture of signs the value of $d\tau^2$ can be positive, negative or zero.
If $d\tau^2$ is less than zero then $d\tau$ must be imaginary, and therefore unphysical. A quick bit of maths will show you that $d\tau^2$ can only be negative if you travel faster than light, and therefore that $c$ is the fastest speed anything in the universe can travel.
So $c$ is special because it determines a fundamental symmetry of the universe.
I've said $c$ is special while Kostya has said the opposite, but actually we are both right.
Kostya is right that there is nothing special about the speed 299,792,458 m/s (though if you change it by much you'll change physics enough that we may not be here :-). However the speed at which light travels is very special because anything travelling at this speed follows a null geodesic, i.e. $d\tau^2 = 0$. This is the sense in I mean that $c$ is special.