An electric field is measured/defined by the intensity of the force (accelerative effect on mass) that is produced by the field. An electric field propagates as a wave. The wave is produced by oscillation (to and fro movement) of an electric charge. So an electric field wave is simply the change over time in the strength of an electric field, produced as a result of the oscillative movement of the charge that produces the field.
Magnetic fields are more complicated, but the analogous distinction exists between magnetic fields and magnetic waves.
How fields themselves work, and their relation to space, time, matter and energy, is the core subject of any unifield theory of physics. For a very good discussion, I recommend Brian Greene's "The Elegant Universe."
Magnetic fields are induced by currents and time-varying electric fields. Waves in an electric field make the field strength vary over time. Time-varying magnetic fields produce electric fields. Waves in the magnetic field make the field strength vary over time. It should be obvious, then, that electric field waves cause magnetic field waves, and magnetic field waves cause electric field waves. It is this process of A causing B causing A by which electromagnetic wave propagation occurs. An electromagnetic wave is both an electric field wave, and also a magnetic field wave. One wave causes the other, recursively.
You can see a graphical depiction here: Animation of the Propagation of Electromagnetic Waves
Ahem: "Unified field theory"
Ahh, this is the core of the matter. I have long been familiar with the graphic image of the magnetic field propagating through space as a sine wave, with the electric field propagating along the same line but polarized at a right angle to the magnetic field. What I was really wondering is, what exactly is the medium through which the wave propagates? Because, as we all know, our conventional understand of waves and energy require that you can't have a wave formed of nothing; the wave is a traveling oscillation jostling gas (or other) molecules around, in the case of sound; or a mobile inequity in the depth of a body of water, in the case of surf.
Apparently, with electromagnetic waves it's some sort of transmissible inequity allowing energy to be stored and released sequentially, in a process that propagates at rate c, in space itself.
I'll have to look up that book.