This is a highly non-trivial issue, mostly because you posted it into the classical-physics forum.
Qualitatively the issue is the following: As you can show using Maxwell's equations and apply it to charged point particles (classical electron theory invented and worked out by H. A. Lorentz in the 1910s after the discovery of the electron and the experimental hints of being a charged point particle by J.J. Thomson), such a particle radiates electromagnetic waves, which carry energy and momentum. Since the total energy and momentum are conserved (due to space-time translation invariance in the special-relativistic framework of Minkowski space), the particle must lose this part of irradiated energy and momentum. This means it must be decelerated due to this energy-momentum loss, and this in turn means that there must be a force that it caused by its own electromagnetic field.
As Lorentz very quickly figured out, there's a lot of trouble in working out this thought mathematically. First of all the self-energy and self-momentum of the electron's own em. field diverges. Even for a point particle at rest, the total energy of its own Coulomb field diverges.
Lorentz cured this by assuming a small but finite extension of the electron. He invented a kind of perturbation theory by first calculating the electromagnetic waves from the accelerated motion of the particle as we all have learned it in our E+M theory lecture (retarded potentials, Lienard-Wiechert fields etc.). Then he considered the backreaction force from the radiation to the particle. He found that part of the force could be lumped into the particle mass. This is quite natural from the point of view of special relativity since any form of energy contributes to the inertia of the particle, and thus also its own em. field does so. This contribution is divergent, when putting the extension of the particle to 0 (i.e., in the limit of a point particle), and Lorentz thus was the first to apply "renormalization", i.e., he put the "bare mass" + "electromagnetic mass" together as the finite physical mass of the electron. The residual force is known as Abraham-Lorentz force (since also Abraham had the same thoughts around the same time).
The trouble with the Abraham Lorentz model is however that one has self-accelerating solutions, which have to be excluded by hand by assuming certain boundary conditions on the solution.
A fully complete self-consistent classical equation of motion for charged point particles is still not found. A very good source on these issues is
Fritz Rohrlich, Classical Charged Particles, World Scientific
Of course, in quantum electrodynamics, this problem is solved to a more satisfactory level, because in the perturbative sense QED is renormalizable, and thus one can calculate the self-energy of the electron to any order of perturbation theory by renormalizing the infinities by lumping them into the unobservable bare mass and wave-function normalization and using the physical values of these quantities (dependent on the renormalization scale and running according to renormalization-group equations), while for the classical theory this is not possible.