Nope. A wave in water or air is the result of the movement of a large number of seperate particles. An electromagnetic wave is NOT this. We call it a wave because during certain observations it ACTS like a wave would.
I have heard of magnetism and relativity, and how that ties together, but people talking about photons confuses me. Do magnets exchange photons? That wouldn't make sense to me. Are the theory that describes magnetism as a relativistic affect and the theory that uses photons different theories?
The carrier of electromagnetic force is the photon. So electric and magentic fields are both propagated via photons. You can think of magnetism as being a consequence of a relativistically transformed electric field but I am of the recollection that not all magnetic fields can be created by transformations from electric fields. In the end, you still need both electric and magnetic fields (or in the case of quantum theory you work with the primitives, the scalar and vector fields, of the electromagnetic field).
The idea of photons is part of the quantum theory of electromagnetics. Quantum electrodynamics using quantum field theory satisfies special relativity. This means that the same relativistic transformations that allow a classical electric field to give rise to a magnetic field in another frame are valid and at work in QFT. So yes, the ideas of photons and transformed fields are compatible since they are present in a common theory.