In Classic Electrodynamics it's well known that an accelerated charge will radiate energy and the radiated power is given by the Larmor Formula
$$P=\frac{\mu_0q^2a^2}{6\pi c},$$
in SI units, where $c$ is the speed of light, $q$ is the charge of the particle and $\mu_0$ the magnetic constant.
Well, to incorporate the Larmor formula into Newtonian mechanics we claim that: if the particle is losing energy while accelerating then the effect can be expressed as if the particle feels the effect of a force opposing its movement. To account for this effect let's say that the 2nd Newton equation can be written as
$$m \vec{\dot{v}}=\vec{F}_\text{ext}+\vec{F}_\text{rad},$$
where $\vec{F}_\text{ext}$ is an exterior force that accelerates the particle and $\vec{F}_\text{rad}$ is some force that would account for the energy (photons) radiated by the accelerated charge.
To continue we impose the following: $\vec{F}_\text{rad}$ has to be such that the work it does is equal to the energy lost by the particle accordingly to the Larmor formula;
We can the use this imposition to write that in an interval $t_2-t_1$, such that the system is in the same state at those instants, I mean the velocity and the acceleration of the particle is the same in $t_1$ and $t_2$ $^{\bf 1}$, the work done by $\vec{F}_\text{rad}$ is equal to the energy lost by the particle. Then:
$$\int_{t_1}^{t_2}\vec{F}_\text{rad}\cdot \vec{v}~dt=-\int_{t_1}^{t_2}\frac{\mu_0q^2}{6\pi c}\dot{\vec{v}}\cdot \dot{\vec{v}}~dt,$$
where the relation $a^2=\dot{\vec{v}}\cdot \dot{\vec{v}}$ was used.
Now, integrating by parts we find:
$$\int_{t_1}^{t_2}\vec{F}_\text{rad}\cdot \vec{v}~dt=\frac{\mu_0q^2}{6\pi c}\int_{t_1}^{t_2}\ddot{\vec{v}}\cdot \vec{v}~dt-\frac{\mu_0q^2}{6\pi c}(\dot{\vec{v}}\cdot \vec{v})|_{t_1}^{t_2}.$$
Since the particle is in the same state in both the instants $t_1$ and $t_2$, the second term of the sum is zero and then we can write:
$$\int_{t_1}^{t_2}\left[ \vec{F}_\text{rad}-\frac{\mu_0q^2}{6\pi c}\dot{\vec{a}} \right] \cdot \vec{v}~dt=0.$$
Since this must be true for all $\vec{v}$ you get the famous Abraham-Lorentz force.:
$$\vec{F}_\text{rad}=\frac{\mu_0q^2}{6\pi c}\dot{\vec{a}}.$$
This equation has some serious problems when you try to use it. You can see those in this reference. Also in that reference Eric Poisson shows the special relativity generalization of the AL force called the Abraham-Lorentz-Dirac force.
As for the second part of the question: It's well known that any interaction that propagates at a finite velocity originates a self-force effect. In Classical mechanics only the electromagnetic interaction travels at finite velocity, $c$, so Newtonian gravity doesn't originate self-force effect since the interaction propagates at infinite velocity.
However, assuming that Lorentz invariance is correct (and at least we physicists believe so, and all the tests indicate towards that belief) nothing can propagate at infinite velocity and there is actually a maximum velocity allowed, $c$, so all interactions should originate a self-force effect (usually called radiation reaction).
PS: If anybody is still with me after this very long answer and you remember me writing that Newtonian gravity wouldn't originate radiation reaction, well in the modern way of looking at gravity (General Relativity) gravitational waves propagate at the speed of light so there must be a gravitational self-force effect... More on that you can check this reference in which Eric Poisson et al. deduce the first order correction to the movement of a particle in curved space-time.
$_{\bf 1}$ More on this you can check the book from griffiths - chapters 10 and 11.
