The upshot is: One part of the field is the Coulomb field "bound" to the particle. It's the electrostatic field it has around it when it is observed at rest in an inertial reference frame. If the charge is moving uniformly, i.e., with constant velocity wrt. an inertial frame, you still have only this field, but you observe both electric and magnetic field components, because of the transformation behavior of the electromagnetic field under Lorentz transformations, which describe how to switch from one inertial frame to another, which is moving with constant velocity against the former.
If now the particle is accelerated, e.g., due to some external electromagnetic field, additional to the (boosted) Coulomb field you also have an electromagnetic radiation field, which transports energy, momentum, and angular momentum to infinity, and this has of course an effect on the motion of the particle. Take the simple case of a particle in an external homogeneous magnetic field. Neglecting the radiation you can solve the equation of motion and get a circle as a trajectory the particle moves along with constant angular velocity (the "cyclotron frequenzy ##\omega=\sqrt{1-\beta^2} qB/m##, where ##\beta=v/c=\text{const}##, and kinetic energy is conserved. However, the particle is obviously accelerated, because the direction of the velocity changes (though its speed stays constant). So the particle radiates electromagnetic waves, and this carries away some of the particle's kinetic energy in form of radiation, which in this case is called cyclotron radiation. That means the particle looses the corresponding amount of kinetic energy irradiated by this radiation. Effectively that has the effect of the "radiation damping". It's similar to friction, only that here the energy is lost to radiation rather than to heat of a medium the particle is moving in.
To take the radiation damping into account you have to calculate the force on the particle due to the interaction with its own electromagnetic field. Doing this naively you get a severe divergence. If you try to calculate this force already for a particle at rest, i.e., its Coulomb field, you hit the corresponding singularity of this field at the position of the particle, but this is of just a physically wrong description, because a particle at rest in an inertial frame should stay at rest, no matter whether it is charged and thus has a Coulomb field around it or not. So you have to subtract this diverging force from its own Coulomb field acting on the particle. This you can do after regularizing the Coulomb field somehow to make it finite at the place of the particle. One strategy already followed by Abraham and Lorentz is physically motivated: You just make the point particle an extended particle. Then the Coulomb field is finite everywhere. As Poincare has stressed first, of course you need some stresses holding the particle together against the repulsive Coulomb force between its parts. Then you can work out an equation of motion, which is however very complicated. You get not a nice local equation of motion but a differential-difference equation due to the retardation effect, i.e., the fact that the electromagnetic interaction propagates at the finite speed of light and not instantaneously. So it takes some tiny time that one part of the extended particle can react to changes of other parts. This retardation effect goes of course away in the limit where you make the extension 0 again, i.e., taking the point-particle limit. One part is of course still diverging, but it can be shown that it is effectively just a contribution to the (invariant) mass of the particle (i.e., the inertia due to the energy contained in the particle's Coulomb field, which of course goes to infinity in the point-particle limit). On the other hand, you cannot observe a charged particle without its Coulomb field, i.e., the infinite contribution to the mass must be lumped together with the unobservable "bare mass" of the particle, together leading to the finite physical mass. After this "mass renormalization" you get an equation of motion, the Lorentz-Abraham-Dirac equation (LAD equation), which is formally finite but pretty strange, because it contains a third-order time derivative of the position, and this causes a lot of trouble like "runaway solutions", where a particle at rest in an inertial frame gets rapidly accelerated "out of nothing", which is of course unphysical, because the particle should stay at rest forever. Another bad feature is "pre-acceleration", i.e., when you switch on an external force, starting accelerating the particle, even when having dropped the unphysical runaway solutions, it turns out that in fact the particle starts to accelerate earlier than the force is switched on. This is clearly violating causality. The best one can finally do is to use the socalled Landau-Lifshitz approximation to the LAD equation, which avoids the pre-acceleration as well as the runaway solutions and is of 2nd order in time as a usual mechanical equation of motion should be and it takes into account the radiation reaction.
The same result you can get much easier with the regularization used by Lechner et al used in the papers quoted above and my SRT FAQ article.
This is of course not a full self-consistent description of the dynamics of a point particle and the electromagnetic field, and so far nobody has come up with a solution for that problem, and it is pretty likely that there is no consistent description, because the notion of a point particle is unphysical. Quantum theory teaches us that you cannot localize a particle definitely, and thus the concept of a point particle doesn't make sense. If you turn to quantum theory, i.e., quantize both the particle and the electromagnetic field, an alternative strategy to derive a classical equation of motion for a "point particle" is to analyze the problem from the point of view of open quantum systems, i.e., you consider the particle and the quantized em. field and consider only the particle. This leads to a non-Markovian quantum Langevin equation for the motion of the particle's average position and momentum, which can be further approximated again by the Landau-Lifshitz approximation of the LAD equation. Of course also in the quantum case you have to renormalize the self-energy of the charged particle as in the classical treatment. For this treatment (though only in the non-relativistic limit), see
G. W. Ford, J. T. Lewis and R. F. O’Connell, Quantum
Langevin equation, Phys. Rev. A 37, 4419 (1988),
https://doi.org/10.1103/PhysRevA.37.4419