DaleSpam has a very good point. The Maxwell equations split into two general sorts, namely in the homogeneous ones, i.e.,
$$\vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0, \quad \vec{\nabla} \cdot \vec{B}=0,$$
which are constraints in addition to the dynamical equations of motion, which are the inhomogeneous Maxwell equations,
$$\vec{\nabla} \times \vec{B} -\frac{1}{c} \partial_t \vec{E}=\frac{1}{c} \vec{j}, \quad \vec{\nabla} \cdot \vec{E}=\rho,$$
where I have used Heaviside-Lorentz (rationalized Gauss) units and wrote down the fundamental Maxwell equations and not the approximate (coarse grained weak-field linear-response) in-medium equations.
The "constraints" let us introduce the scalar and vector potential (or relativistically the four-vector potential), which however introduces gauge invariance, which makes the explicit causility structure a bit hidden, depending on the choice of gauge, which is arbitrary and made to simplify the problem you want to solve.
In classical electromagnetism, however we don't need to introduce the potentials but work directly with the electromagnetic field, i.e., ##\vec{E}## and ##\vec{B}## which are gauge invariant and observable quantities, i.e., they should be interpretible in terms of causal equations of motion, and indeed they are.
The first thing to observe is that the Maxwell equations put a constraint on the possible charge-current-density distributions, which is also a consequence of the underlying gauge invariance. Taking the divergence of the Ampere-Maxwell law gives
$$-\frac{1}{c} \partial_t \vec{\nabla} \cdot \vec{E}=\frac{1}{c} \vec{\nabla} \cdot \vec{j}.$$
Now with Gauß's Law we get
$$\partial_t \rho+\vec{\nabla} \cdot \vec{j}=0,$$
i.e., even without the equations of motion for the charged particles we know that the electric charge must be strictly conserved, i.e., you have a constraint for the equations of motion of the charges. There must be a (global) symmetry leading to the conservation of the charge you couple the electromagnetic field to. That's a hint of the necessity of gauge invariance of electromagnetism which results from the analysis of the group structure of the proper orthochronous Poincare group, but that's another story.
The next step is to decouple the equations for ##\vec{E}## and ##\vec{B}## with help of the homogeneous Maxwell equations. To this end it's clear that we should try it with the curl of the Ampere Maxwell Law in order to use Faraday's Law to get rid of ##\vec{E}##:
$$\vec{\nabla} \times (\vec{\nabla} \times \vec{B})-\frac{1}{c} \partial_t \vec{\nabla} \times \vec{E}=\frac{1}{c} \vec{\nabla} \times \vec{j},$$
and indeed with Faraday's Law we have
$$\vec{\nabla} \times (\vec{\nabla} \times \vec{B}) + \frac{1}{c}^2 \partial_t^2 \vec{B}=\frac{1}{c} \vec{\nabla} \times \vec{j}.$$
Now we work in Cartesian coordinates, and then we can simplify the "double curl" and also use the other homogeneous Maxwell equation (Gauß's Law for the magnetic field, saying that there are no magnetic monopoles):
$$\left (\frac{1}{c^2} \partial_t^2 - \Delta \right) \vec{B} =\Box \vec{B}=\frac{1}{c} \vec{\nabla} \times \vec{j}.$$
Now it is clear that the solution of this wave equation is only unique when we invoke the assumption of causility. As any linear differential equation it consists of the full solution of the homogeneous equation, whose concrete form is determined by the initial conditions given by ##\vec{B}## and ##\partial_t \vec{B}## at a time ##t_0##, and the inhomogeneous solution, which relates the sources of the fields, ##\rho## and ##\vec{j}## to the fields and which should obey the causility constraint, i.e., the fields at ##t## should be a functional of the sources at earlier times ##t'<t##. Technically this determines that of all Green's functions of the d'Alembert operator ##\Box## we choose the retarded one, which is very simple since we deal with massless fields and operate in three spactial dimension, where the naive Huygens principle is indeed the correct answer:
$$\Delta_{\text{ret}}(t,\vec{x})=\frac{1}{4 \pi |\vec{x}|} \Theta(t) \delta(t-|\vec{x}|/c).$$
Thus the inhomogeneous part (which is the solution for the case where the sources where adiabatically switched on in the remote past) is given by
$$\vec{B}(t,\vec{x})=\int_{\mathbb{R}^4} \mathrm{d}^4 x' \Delta_{\text{ret}}(t-t',\vec{x}-\vec{x}') \frac{1}{c} \vec{\nabla}' \times \vec{j}(t',\vec{x}').$$
In the same way we get a wave equation for ##\vec{E}## by taking the time derivative of the Ampere-Maxwell law and eliminate ##\vec{B}## with help of Faraday's Law:
$$\frac{1}{c} \vec{\nabla} \times \partial_t \vec{B}-\frac{1}{c^2} \partial_t^2 \vec{E}=\frac{1}{c^2} \partial_t \vec{j},$$
$$-\vec{\nabla} \times (\vec{\nabla} \times \vec{E})-\frac{1}{c^2} \partial_t^2 \vec{E}=\frac{1}{c^2} \partial_t \vec{j}.$$
Again resolving the double curl for Cartesian components and using Gauß's Law for the electric field gives
$$\Box \vec{E}=\vec{\nabla} \rho -\frac{1}{c^2} \partial_t \vec{j}.$$
The causal solution is again given with help of the retarded propagator.
These two retarded solutions can be simplified by integrating out the ##\delta## distributions, where one must be a bit careful with the derivatives of ##\rho## and ##\vec{j}## occurring in the integrands. The resulting equations are, for some strange reason, named Jefimenko equations, although the retarded solutions of the Maxwell equations reach back to the mid 18's by Lorenz (the Danish physicist, not the Dutch Lorentz):
http://en.wikipedia.org/wiki/Jefimenko's_equations
For our argument, the somewhat simpler looking forms with the ##\delta## distribution in the integrals make however clear the conceptual structure behind the Maxwell equations: The homogeneous ones are constraints while the inhomogeneous ones are the equations of motion connecting the sources, ##\rho## and ##\vec{j}## causally to the corresponding "field response".
You can of course also find other equations, connecting, e.g., the displacement current as a part of the sources of the magnetic field, but as soon as you want the full solution of the problem to derive the electromagnetic field from given charge-current distributions, you'll see that this point of view is utmost more complicated (and I dare say even "ugly"), because you get a pretty nonlocal way to express the magnetic field with the displacement current as a source, which also hides the beautiful "causal structure" of the Jefimenko expressions, which are conceptually much cleaner and last but not least much more in the spirit of relativistic field theory.
