What defines the mass of elementary particle? The electron is particle.
The mass of electron is $9.10938215(45)\times 10^{−31}\, {\rm kg}$.
But why is the mass exactly what it is?
What in physics defines the mass of elementary particle?
 A: Right now, the elementary particle masses simply have to be measured. We don't have any explanation for why the masses have the particular values they do.
One of the major goals of unified theories (string theory and the like) is to be able to derive the elementary particle masses, or at least derive the ratios between the masses of different particles. But we're a long way off from doing so.
A: Dear GJ, the mass of every electron is exactly the same - as long as our experiments and theories may say - and the same value of mass manifests itself 


*

*in the energy/mass conservation law; 

*in the source of the gravitational field that the object containing massive particles such as the electron produces; 

*in the resistance towards acceleration that the massive objects have.
All those versions of the "mass" are the same quantity.
In our modern theories of the world, the mass of the electron is the coefficient of a "mass term" in the Lagrangian or Hamiltonian, $-m\bar\psi\psi$, that is almost fundamental and can't have any deeper explanation.
Well, almost. In fact, in the "electroweak theory" - unifying all electromagnetic phenomena (including light) with the weak nuclear interactions (notorious as the origin of the beta-decay) - the mass of electrons (and other particles) arises from the interactions with the Higgs field. So the actual term is actually
$$-y h \bar\psi\psi$$
where $\psi$ is the Dirac field for the electron. Here, $h$ is the Higgs field which takes the nonzero value of $h=v$ in the vacuum - the spontaneous symmetry breaking (Higgs mechanism). This value of $v$, about $247$ GeV, is universal for all particles. However, the difficult information about the electron mass is moved to the parameter $y$, the so-called Yukawa coupling, which is dimensionless (no units) and whose value is much smaller than one.
When the Universe is described by string theory, the value of the Yukawa coupling $y$ and the Higgs value in the vacuum, $v$, can be in principle calculated from first principles (in the natural units, e.g. the Planck units, not in kilograms - the unit of "kilogram" was a randomly chosen social convention that the farmers chose to sell vegetables, without any justification why it's not e.g. $1.3$ times different).
However, there are many (but discretely) possible Universes according to string theory and we can't say which one is right, so in practice, we can't calculate the value of the electron mass from purely theoretical considerations. So far, we need to measure it to know the value. Chances are that the situation won't change for quite some time.
A: A not standard answer, i.e. you should not use it in scholar context:  
The book of Douglas Pinnow ' (1) The Resonant Universe' is a cheap monography of his model of particles.
It is entirely based in Electromagnetics, and has only one parameter ($m_e$- electron mass) and derives all the relevant particle properties, to the laymen and also with math, well within 1%  for barionic masses without the barionic 'spin crisis'.
Concepts like mass, gravitation and charge become visually appealing to my mind. 
He predicted a new particle, $Z_0$ like, near 780 GeV, and the next one near 1450 GeV. As SM is probably in trouble, as seen in recent experimental findings, may be we should read the book.  
There is no need of weak or strong force. Only EM, he followed after Kaluza-Klein, Goedecke, Haus (non-radiating condition EM sources). 
Also by copy_paste ;-) "In addition to generating the correct magnetic moments of the proton and neutron without adjustable parameters, such as required by the standard model, the resonant model predicts a negative value for the magnetic moment of the neutral sigma baryon wich has yet to be measured".The SM predicts a positive value.  
It seems that I'm the only reader of his book. 
(1) - He had a long career in nuclear physics and light engeneering.
