That question, however, cannot be answered at this level. You need some math to express relativistic QFT correctly. The worst thing is to tell people photons where something like particles. Forget about QFT, before you have learnt classical field theory.
The idea of fields was one of the most important thoughts in physics. The problem was to understand, how far distant objects interact. The first theory of such an interaction in modern physics (i.e., in the physics that has been discovered since about 1600) was Newton's theory of the gravitational interaction. From thinking about the known laws of motion of planets around the Sun and the Moon running around the Earth Newton came to the conclusion that there must be "an action at a distance" between two massive bodies (which can be idealized to point masses), i.e., that there's a force proportional to the two masses, to the inverse of the distance between the bodies squared and directed along the straight line between the bodies (and being attractive). Already Newton had his quibbles about this "action at a distance", but he thought that the fact that indeed this assumption of an force acting instantaneously must be right, because he could derive Kepler's Laws of motion from this assumption about the gravitational force, and so it must be right, and thus he didn't want to introduce "further hypotheses" (that was the place, where he wrote this famous statement "hyptheses non fingo").
Now it took almost 2 centuries until Faraday after years of careful experimenting with electromagnetism came to the conclusion that in fact the elecromagnetic interactions between charged particles can and should rather be described by a more local picture, i.e., that the force ##\vec{F}=q (\vec{E}+\vec{v} \times \vec{B})## acts on the particle not due to some action at a distance with other charged (and moving) particles but that this force in fact is due to the presence of the electric and magnetic fields being present at any time and at all positions of the particle. This is the concept of "local interactions", i.e., that with its charge the particles have their electromagnetic field around them and then this field leads to the force acting on another charged particle.
This also indicates that there should be physical laws that describes how these fields look given the charges and their motion. This has been answered by Maxwell a few years later by using Faraday's careful descriptions of his observations how these fields should behave.
Both Maxwell and Faraday, couldn't know that this field concept in fact would become very important as a genral paradigm in the mathematical description of Nature, but Maxwell's equations lead in the late 19th and the beginning 20th century to the development of the theory of relativity. First came the special theory of relativity, which has been formulated at the same time by Poincare and Einstein with Einstein having the most consistent physical interpretation of it. According to Einstein (1905) the problem was that Maxwell's equations did not fit to the established theory of space and time due to Galilei and Newton, i.e., to keep the well-established independence of the natural laws on the inertial frame of reference you use to formulate them, one had to change the very description of space and time itself to make this valid for Maxwell's equations too. This implied in fact that the speed of light (i.e., electromagnetic waves) must be independent of the light-sources velocity with respect to any inertial reference frame, and this implied that the speed of light is the largest speed with which causal effects can propagate, and that was precisely what Maxwell's equations said for the electromagnetic field: The value of these fields depend only on the charge and current distributions such that a change in these distributions at some place can only have an effect on the fields in a distant place after a time it takes for light to go from the sources' place to the place you observe the fields.
Given this "retardation effect" the fields also save the validity of the basic conservation laws: Because it takes some time if one charge moves to take effect on the motion of another far distant charge, Newton's 3rd Law cannot be true anymore in its original sense with instantaneous actions at a distance and thus it seems as if also the conservation law for momentum become invalid, but that's not the case, if you interpret also the fields as some dynamical entity, which indeed they are due to Maxwell's equations, which indeed turned out to be consistent with the new spacetime model introduced by Einstein (and Minkowski for the full understanding of the underlying math), the fields also carry momentum (as well as energy and angular momentum), and then there's no problem anymore with the finite propagation of actions and the conservation of momentum, because if a charge is accelerated this changes its electromagnetic field in such a way that the field takes the corresponding change of momentum, and this is due to the charge being acclerated at its place, i.e., nothing needs to go faster than the speed of light to keep total momentum always conserved, because it can be conserved locally due to the presence of the fields.
The electromagnetic from this modern point of view is a fundamental entity, which cannot be explained by more fundamental things. It's part of the dynamics as the particles are fundamental entities taking part in the dynamics.
Now it's clear, that (according to this modern description of the electromagnetic interaction) the compass needle feels a torque due to the presence of the Earth's magnetic field, which itself is due to (afaik still not fully understood) motions of the charged fluid in the Earth's interior:
https://en.wikipedia.org/wiki/Dynamo_theory
Now this is however not the end of the story, because there was another "revolution" in physics in 1925, quantum theory. This lead to a reconception of how to describe "particles and fields". The most comprehensive theory we have today about "elementary particles" and their "interactions/fields" indeed is entirely a quantum-field theoretical description, and this can not really be described properly without using quite formal mathematics. The upshot is that relativistic quantum field theory solves many problems the above classical modern theory has and which I simply didn't mention.
One of the problems that could not be solved within classical theory of particles and fields itself can be understood to some extent without math: It's known as the problem of "radiation reaction". It has to do with the solution of the problem with energy and momentum conservation in a theory where instantaneous interactions between particles cannot be used. As explained above, the solution is that the energy and momentum change of a charge due to it being accelerated by some force, was to assume that the field is a dynamical entity itself being able to exchange energy and momentum with the particle at the position of the particle. But now one also has to think what this means for the particle! It looses energy and momentum to the field, and in the case of the electromagnetic field that energy and momentum manifests itself as an electromagnetic wave, which carries this energy and momentum away from the particle. Looking on the particle that means that this loss of energy and momentum is as if there's a "friction force" acting on the particle, but it's not a friction force between particles, which manifests as heat, but as electromagnetic radiation. So to understand this transfer of energy and momentum from the particle to the field you have to think about the interaction of the particle with its own field, but that's a big trouble if you work with point particles. As the Coulomb field of a static particle also the full electromagnetic field of a moving, accelerated particle becomes infinite at the place of this particle, and the energy and momentum diverge. This can be solved partially by some mathematical tricks. One of the infinities can be lumped into the mass of the particle and then setting the total mass of the particle (i.e., it's "bare mass" it would have even if there were no electromagnetic field around it and its "electromagnetic mass" added together) to the finite observed mass of the particle. This results, however in an equation of motion for the particle (known as "Abraham-Lorentz-Dirac equation" or "LAD equation") which brings further trouble, violating causality, and it can only be fully "repaired" by considering an approximation, known as the "Landau-Lifshitz approximation", and that's how far you can get within classical physics about a self-consistent dynamical description of point particles and electromagnetic fields.
In the quantized theory you describe also the particles by (quantized) fields, and their the "infinities" also occur when considering the self-interactions between the particle with its own electromagnetic field, but it can be cured by lumping infinities into unobservable masses and charges such that the observable masses and charges are finite and can be given their observed values.
Quantum field theory thus describes both particles and the electromagnetic field successfully on the same footing, and this implies that what you classicall consider as a field also describes phenomena which classically are similar to properties of particles. It's also the other way round: Also things which we consider as particles in the classical theory (e.g., electrons) are in QFT described by (quantized) fields and also there are phenomena which in classical theory would be regarded as typical field phenomena. One example is the famous double-slit experiment, where you shoot electrons (one at a time!) through a double slit. Each electron makes a point on a photo plate ("particle like feature", i.e., the electron hits only one point on the plate) but looking at the distribution of many electons shot through the slits looks as if a wave goes through the slits, i.e., yous see an interence pattern as if you do the experiment with light, where the partial waves coming from the two slits interfere on the screen resulting in a diffraction pattern.
On the other hand also the electromagnetic field can manifest itself in phenomena looking more particle than field/wave like. That's how the necessity of "field quantization" in fact has been discoved by Planck in 1900. To understand the spectrum of the electromagnetic radiation coming from a hot body ("thermal radiation") Planck had to assume that an electromagnetic wave with frequency ##f## can exchange only discrete lumps of energy of the size ##E=h f##, where ##h## (Planck's quantum of action) was a newly introduced fundamental constant of Nature. This in turns lead Einstein to the assumption that light must have besides the long accepted wave properties some particle features. This somewhat inconsistent picture, known as "wave-particle dualism" has been resolved by modern quantum (field) theory discovered in 1925/26 in various forms by Born, Jordan, and Heisenberg ("matrix mechanics"), Schrödinger ("wave mechanics"), and Dirac ("transformation theory").
Now modern QFT, however, tells us that the idea with "particles" and "fields" has not to be taken too far. E.g., the quantum of the electromagnetic field, called "photon", does not admit the definition of a "position observable" in the literal sense as it is possible for massive "particles", i.e., a "photon" cannot be localized. It is much better thought of in a non-mathematical way, as a very specific form of an electromagnetic wave. If this wave has a frequency of ##f## and a wave vector ##\vec{k}## it carries an energy ##E=h \nu## and a momentum of ##\vec{p}=\hbar \vec{k}=h \vec{k}/(2 \pi)##, and when it hits a photo plate it blackens only one grain on this plate and is aborbed in this process giving its entire energy and momentum to the plate. In this sense it has some kind of "particle property", but you cannot in any clear sense say there has been a particle at a certain place before the photon made the black spot on the photoplate. Also it is impossible to predict, at which point the photon will make the spot. All QFT tells you is the probability that the photon hits the photo plate at any position.
Also the idea that classical electromagnetic fields like the magnetic field of the Earth or the light from the Sun or light emitted by a laser pointer etc. as some "stream of photons" is misleading. Such "macroscopic" electromagnetic fields have not even a well-determined number of photons but the probability to detect a number of photons at a given place is Poisson-distributed. From a QFT point of view such fields are called "coherent states" of the electromagnetic quantum fields, but this I really cannot in any way adequately describe without math.
