What is an electric field? I know that we can define electric fields by the influence they have on certain kinds of matter (in particular, the charged kind), and that they store potential energy, but my question is what actually IS an electric field? We understand that a gravitational field is actually the curvature of spacetime (whatever spacetime actually is) in the vicinity of matter, but is there a similar explanation for what an electric field is, or what charge is? Are they just labels we gave to some observed phenomena that stand in a mathematical relationship, or do we understand something more about the physical nature of what the field actually is?   
 A: The EM field is a physical field created by electrically charged objects. It extends indefinitely through space.
You are asking about the electric field but we talk about EM field, and the electric field and the magnetic field can be observer dependent. A magnetic field can be viewed by one observer as electric field and by another one as magnetic field.
Now you are asking if what it is. The EM field is fundamental, and as you say it is a phenomenon, and in our currently accepted theories, the SM, QM and QFT, we describe this phenomenon with mathematical models, that best fit the data from the experiments.
When we try to describe the EM field as it interacts with something, we use virtual photons. Virtual photons mediate the EM field's interactions. These are not real photons. Virtual photons are just a mathematical model, that best describe the interaction between the EM field and something that it interacts with. In reality we do not know what it is, how it works really, we do not know what really mediates these interaction, we do not really know how they interact. What we do know is the data from experiments, and we built up mathematical models, called virtual photons to best describe these interactions.
Now you are asking whether the EM field can be described as the gravitational field when we use the phrase "gravity bends spacetime".
Yes there is similarity. But, gravity is always:


*

*attractive

*interacts with all known particles in the SM
So as per GR, and the SM, all known particles interact with the gravitational field. Thus, we can say that the gravitational field has an effect on spacetime, so that the area of space where the gravitational field exists, will have an effect on all known particles in the SM, so that these particles will have an altered trajectory when they interact with the gravitational field. 
Because gravity is always attractive, the particles will always bend one way, towards the center of mass. And because gravity acts on all known particles in the SM, we tend to use this phrase "gravity bends spacetime". In reality we do not know what really bends, we do not know how it bends, and we do not know if anything really bends. All we know is that the particles will have an altered trajectory in all experiments.
Now with the EM field:


*

*sometimes attractive, sometimes repulsive

*acts on only some particles in the SM
What we can say, is that the EM field has an effect on spacetime, so that the area of spacetime, where the EM field exists, will have an effect on some particles (that interact electromagnetically) so that these particles will have an altered trajectory when they interact with the EM field.
But because the EM field is sometimes attractive, sometimes repulsive, and it only acts on some particles in the SM, we cannot say that " the EM field bends spacetime". In reality we do not know if anything bends, all we know is that some particles will have altered trajectory in all experiments.
A: In biology, animals are divided into species and usually you don't ask what a cat or dog is. But you are able to describe them.
The same is in physics. An electron has a charge and we divide charged  subatomic particles into species with exactly two opposite charges.
On the other side it was introduced the electric field. For two electrons the electric field between them and their electric charges are synonyms in the sense, that their charges are defined through the force they exercise on each other. More precisely, our knowledge about charges comes from the effecting electric field between charges. It’s simply avoiding to characterize an electron as “the particle with a negative electric field” instead of the electron is “a negative charge”.

I know that we can define electric fields by the influence they have on certain kinds of matter (in particular, the charged kind)...

is in principle the same what I've explained in so many words.

... but my question is what actually IS an electric field? ... Are they just labels we gave to some observed phenomena that stand in a mathematical relationship, or do we understand something more about the physical nature of what the field actually is?

Good point. The introduction of the field was so successful that so far there was no urge to investigate the consistency of the field. For the gravitational field the constituent of the field was introduced as the graviton. “...the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravity.” For electric fields such an interpretation attempt is missing. We are satisfied with what we have. Our thirst for research in this area is currently too low.
A: Electric field is region of space in which if any test charge comes will experience a electrostatic force this defination or hypothesis was given by michael faraday to explain how the charged bodies interact.But this is hypothesis till  this date.Nobody knows there really exsists electric field thats why we try to explain it by quantam mechanics that the charged particle exchange photon particle thus exerting a electrostatic force.But electric field allows us to understand and explain certain things very easily thats why we use it.In physics we cannot know always why something happens.eg why F=dp/dt
A: The concept of a field (electric, magnetic, gravitational) was put forth to explain observable phenomena in which one body could impose a force on another body, without there being physical contact between the bodies. In particular, the presence of a force without physical contact was described “action at a distance”, a concept that troubled physicists going back to Newton. Newton said 
"It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual contact".
I believe it was Michael Faraday who suggested that the "something else" that was not "material" that enabled matter to exert a force on other matter without contact was a field. 
I'm sure you’ve seen iron filings used to visually demonstrate the presence of the magnetic field of a bar magnet, in which the filings align themselves with the magnetic field.
Hope this helps.
