The electromagnetic interaction affects charged objects -- that's pretty much what we mean by charge, in that it is the stuff that is sensitive to the presence of electric and magnetic fields. Charged objects also create static electric fields which are responsible for the attraction and repulsion of other charged objects. When in motion, charged objects are responsible for the creation of magnetic fields too. When accelerated, charged objects will radiate electromagnetic waves.
The quantum picture of the electromagnetic interaction is the exchange of photons between charged particles. We say that the photon field "couples" to the fields of charged particles, such as electrons and quarks, where the word "couple" comes from the language of coupled oscillators exchanging energy. Photons themselve are not charged, but they are only allowed to interact which charged particles. This means that photons do not interact with each other (except rarely in processes involving exchange of "virtual" charged particles). The strong nuclear force is a lot like the electromagnetic force except that the gluons (which play a similar role as the photon) do have strong nuclear force charge (whimsically called "color"), and therefore do interact with each other. This self-interaction produces a sticky mess which binds the quarks together inside of hadrons and does not let them escape individually. I am very glad that we have a variety of forces -- electromagnetic, strong, weak, and gravitational, because they are all needed to make the universe work the way it does.
On your second question, it was in fact believed for a long time that a medium was required for electromagnetic wave propagation. But starting in the late 19th century, with the experiment of Michelson and Morley, which was designed to measure the Earth's velocity with respect to this medium, it has been rejected as a possible model of electricity and magnetism. The observation of that experiment and subsequent ones is that light travels at the same speed in all directions, and more surprisingly, according to any observer who is moving relative to any other observer with any possible speed. Sound travels along with its medium. If the wind blows, then the velocity of sound adds to the velocity of the wind. Making a model in which light travels at the same speed according to all observers required abandoning our traditional ideas of wave propagation in a medium, and also the ideas of uniquely defined space and time coordinates that all observers can agree on.
Who knows? Maybe there are "ethers" out there which fill up all of space. But we don't need to invoke them in our models in order to explain anything -- they just make life more complicated without predicting anything. And in the case of light propagation, they get downright misleading because proposing a medium in which light propgates immediately raises the question of preferred reference frames (like the one moving with the medium) which conflicts with observation.
Just a note to elaborate on what Tom wrote: At some point in our description of things, we have to get to some bedrock mathematical ingredients which we cannot describe in terms of some deeper level. Right now, the basic fields like E-M are that level. If there were some description of some stringy space-time (for example) of which the E-M waves were excitations, then we would still be stuck, unable to answer what the mathematical features of that space-time were 'made of'. Mike W.
(published on 10/22/2007)