Is there a more elementary example of the holographic principle? Someone was telling me about the holographic principle, basically he said that the state of a system is determined entirely by the values of various physical quantities on its boundary. This is not exactly what it says on Wikipedia's Holographic Principle article - "The holographic principle is a principle of string theories ... that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region—preferably a light-like boundary like a gravitational horizon".
I think Wikipedia's definition is a bit confusing to a beginner, since string theories have maybe 10 dimensions, so it's not clear what "lower-dimensional" means (9 dimensions?), and also since it requires one to understand the concept of a "light-like" geodesic and how this concept can be extended to define a "light-like boundary" surface.
I am wondering if the same principle can be stated in a more elementary form without invoking string theory. For example, is it the case that knowing the electric and magnetic field on the surface of a sphere will tell me the spatial distribution and velocities of charges within the sphere? If so, does this simpler idea have a name? (Would it be incorrect to refer to it as the holographic principle?)
 A: Disclaimer: I'm not an expert on the holographic principle. I'm posting this answer because it might have some limited value, but I hope somebody else steps in to give you a real answer.

is it the case that knowing the electric and magnetic field on the surface of a sphere will tell me the spatial distribution and velocities of charges within the sphere?

No. 
For a counterexample, consider three concentric shells: The innermost one has static electric charge $+Q$ spread uniformly over its surface, the middle one has charge $-Q$ spread uniformly over its surface, and the outer one is where you make your observations. According to classical electrodynamics, the electric and magnetic fields on the outer surface are both zero for any $Q$, so value of $Q$ is not encoded on the boundary.
The holographic principle is different, and despite the name, it's also different than an ordinary hologram. In an ordinary hologram of an opaque object, you don't see the inside of the object. In the thing called the holographic principle, the lower-dimensional encoding is all-seeing, and the possibility of such an all-seeing lower-dimensional encoding is closely associated with impossibility of cramming unlimited amounts of information into arbitrarily small spaces in the bulk. That limitation, in turn, is closely related to the fact that massive objects automatically bend spacetime, the phenomenon we know as gravity. 
(Caveat: the question of whether or not a "holographic screen" can encode what's inside a black hole might still be unsettled, but the question of whether or not a black hole really has an "inside" that's informationally independent from its "outside" might also still be unsettled. ...or maybe they are settled and I just haven't learned how yet. I have a lot to learn.)

Is there a more elementary example of the holographic principle?

The the AdS/CFT correspondence is most well-developed family of examples we have that exhibit the holographic principle, but even the simplest examples of the AdS/CFT correspondence (like AdS$_3$/CFT$_2$) are still far from simple by my standards. If a more accessible example of the holographic principle does exist, I hope that somebody else posts an answer about it, because I'd love to learn about it.
A: In the holographic principle for quantum gravity the DOFs are encoded on a spacetime codimension-2 Cauchy surface. A conventional hyperbolic system has a spacetime codimension-1 Cauchy surface. This is e.g. the case for E&M and Maxwell's equations, cf. e.g. my Phys.SE answer here. Similarly, Dirichlet problems are specified on a codimension-1 boundary. See also this related Phys.SE post.
