How do fields work? By fields, I am referring to gravitational, electric... Basically, how can two things interact at a distance? I know that we have mathematical descriptions of the phenomenon, but, physically speaking, I don't get it. Do particles, somehow, emit some kind of bullets, continually and in all directions that cause other particles to move, and, also somehow, one's "bullets" do not hit another's? ...
 A: 
By fields, I am referring to gravitational, electric... Basically, how can two things interact at a distance? I know that we have mathematical descriptions of the phenomenon, but, physically speaking, I don't get it.

Physics is the discipline that describes nature using mathematical models , and assumptions that are axiomatic, taken from observations , called laws,principles, postulates. Axiomatic means that "it is what has been observed". The fields you describe are classical fields that arise in the mathematic fit of the data and  are chosen because the mathematics is also predictive of new boundary condition solutions. 
Fields arise because of the mathematical modeling of observations.
The fields you describe are classical fields.

Do particles, somehow, emit some kind of bullets, continually and in all directions that cause other particles to move, and, also somehow, one's "bullets" do not hit another's? ...

This is interesting because you are describing by words the complex mathematics used in quantum mechanics, where the classical fields can be explained by the exchange of virtual photons between interacting charged particles.
Here is how the interaction between two electrons is described in quantum field theory:

they exchange a "virtual" photon, another mathematical representation, carrying the quantum numbers, but not the mass of the photon. 
The interesting thing is that this mathematical exchange of virtual photons can be shown to build up the classical electric field .
So in a pictorial mathematical sense, yes, a lot of virtual photons are playing ball, and the balls  do not interact with each other because photon photon interactions are very improbable even for real photons, and more so for virtual ones.
A: I take issue with the implication that the mathematical description is different from the physical picture. In describing Newtonian mechanics, we have the great advantage of being able to visualize colliding balls and vibrating springs. In describing special relativity, we have the luxury of putting Alice and Bob on spaceships. The demand for closer-to-home analogies is a natural one, and often helps gain an insight into the dynamics of a system. The problem is, in order to have a good analogy when describing a phenomenon, our language must have great scope and flexibility to understand all the aspects of a physical situation. The one language that is unfailingly flexible enough, and general enough to do this, is mathematics. 
The premise of physics is simple, observe nature, and build descriptive, as well as predictive models. Mathematics is just a better language for making analogies than balls, bullets and masses moving on stretched membranes (general relativity), in that it breaks down far less often than analogies involving the latter. This is especially so when trying to tie classical intuition up with quantum phenomena. For example, take the case of wave-particle duality, where our two standard classical pictures both come up short individually. So it's natural to ask the question, what does it all mean in terms of ...insert favourite language of analogies..., but chances are, it's going to be a less helpful description than a description in terms of mathematics. 
Action at a distance, gravitational and electric fields, and other quantum interactions we observe really have much more similarity with linear algebra, calculus of variations and integrations over infinite dimensional configuration spaces than they have with other languages we might use to describe them. Therefore, I would argue that if someone claims that interaction at a distance is whatever an equation says it is, then that is the most elegant physical description of what is going on. 
A: There is a field for each fundamental particle. These fields are present throughout the entire universe. One way to think of it might be to think of water, but don’t visualize being on the water’s surface. Think of being in the water below the surface. If you disturb the water by moving your hand the “water field” will vibrate. This vibration wave,excitation, will propagate out away from you. It will have the ability to affect things that are at a distance from you. You should know; however, that exciting or disturbing the field requires a certain amount of energy. Each field has its own minimum amounts. Once enough energy (in this example your waving hand) is put into the field a wave is generated that propagates out as a wave in the field. This energy wave has the potential to be a particle, because it has energy and energy and mass are linked. 
So the take away on this is the fields are already in place. Nothing needs to travel through empty space because there is no empty space. It’s all fields that can be disturbed with a minimum amount of energy. This energy has the potential to be a particle if measured. I am not a physicist so this is a layman’s answer. I hope it helps.
