Right now, this is what I'm thinking. First, what do we mean by 'traveling' and 'accelerating'? This brings us back to the concept of inertial reference frames and the relativity of motion. (Wasn't Einstein's first paper on special relativity titled "On the electrodynamics of moving bodies", or some such? ) Suppose you and an electric charge are in a reference frame within which the charge is not moving. If you are close enough to the charge, you could measure the strength of an electric field around the particle with an electrometer or some such. You could measure the distance dependence of the strength and direction of the force exerted by that field on a test charge. As expected, the force will fall off as the inverse square of the distance, and the direction of the force will be either towards or away from the charge, i.e. the force behaves like a vector, with magnitude and direction. Mathematically speaking, there is what's called a 'vector potential' - a mathematical function that assigns a vector to every point in space. But that field can't extend to the edge of the universe unless it sat there immobile in your frame since the beginning of time. If it hasn't, then there will be an edge to that field, a moving boundary (again in your reference frame). Another measuring instrument located beyond this boundary will detect something like a tsunami, an abrupt increase in the electric field that remains constant after the event. In other words, it will have detected a wave (sort of).
Now let the charge in your laboratory begin moving. Both the distance and direction of the charge w.r.t. (with respect to) your lab instruments will change and eventually the same thing will be detected in any reference frame anywhere ( as long as it isn't traveling together with the charge, the case if the charge is constant w.r.t. that other frame.) as a wave-front passing over the instruments. Your moving charge has just created a wave in your lab. That wave will then propagate throughout space. Until it reaches other detectors, however, the 'old' field will be measured there, with the same vector field as before. But the charge that created it is no longer at its center, i.e. the field out there is no longer in step with the charge in your lab and in this sense, it exists without the charge that created it, since that charge is no longer 'there'! Suppose now the charge in question moves sinusoidally. The perturbation in the electric field too will oscillate sinusoidally. Your instruments will have detected an EM wave, and as such this oscillating field will spread throughout space and eventually be detected in all reference frames out there as a classical EM wave.
What I have just said is wrong. Why? I called a purely electric field perturbation an EM wave. What about the 'M' in EM? I have omitted a very important pair of concepts here: acceleration and magnetism. Maxwell's field equations show that when electric charges move they create magnetic fields, and vice versa. Furthermore, when charges accelerate, perturbations of their associated magnetic fields propagate in tandem with the perturbations of their electric fields. Hence, we have a true EM wave. I never studied Maxwell's equations and special relativity, so I'm slipping into waters over my head, so I'll stop except to say that this has been a purely classical explanation. It doesn't take into account quantum effects, i.e. those photons, or relativity which shows that EM radiation doesn't travel as a nice centrosymmetric wave. This is important in understanding synchrotron particle accelerators, where particles traveling in circles close to the speed of light constantly emit x rays concentrated in the instantaneous direction of the particles. The accelerator must constantly replenish the lost energy and add even more if it wants to accelerate the particles even further. Furthermore, the intense, focused beam of x-rays is useful in its own right, for experiments involving materials structure analysis, and so forth.
There's also an important difference between EM and gravity fields. Gravity fields and waves carry only one force, gravity, not 2 as in EM waves. I imagine this was a bete noir in Einstein's later days, when he was trying to create a field theory that would unify EM and gravity.
