A field is a mathematical object which assigns a value to every point in space or spacetime. The value can be a number, or it can be a vector or tensor depending on the kind of field. The electromagnetic field is a matrix at each point, but it can conveniently be rewritten as two vectors (for the electric field and the magnetic field). It's just pure math. There isn't any sort of "aether" that the field rests in.
AFAIK, the EM field does not have to have quantized parameters (like angular momentum), but any measurements of the field will give quantized values. So, we imagine that the field is made up of lumps called photons. But the thing is, not all aspects of the EM field can assigned to the lumps. So, we invent something called virtual photons which don't follow all the rules of photons. (In particular, they can have energy that is not proportional to the momentum.) But they do follow other rules, such as some interaction cross sections, so it makes sense to treat them like particles in some respects. Both particles and virtual particles are somewhat artificial things, when the most fundamental object is the field. But the thing is, you can't measure the field directly. You can only measure the particles, and the effects that virtual particles have on particles.
When two electrons approach each other, they don't fire off virtual photons. Rather, they interact with the EM field. And the part of the field they interact with hasn't been associated with any real photons, so we say it interacted with virtual photons. Actually, it is the electron field that interacts with the electromagnetic field.
It was a general comment, not addressed directly to you. Explanations via virtual particles are quite good and easy to grasp (that's why they are widely used in pop-sci books), but you always have to remember that they are not completely "true" (whatever that word means).