What is electron? In quantum mechanics, we easily talk about some "particles" or "somethings" like electron and photon. Besides, in classical mechanics we talk about particles that have mass. As we know, one can formulate axiomatically the meaning of a particle in Newtonian mechanics. Is there any similiar approach to understand "what is electron" in quantum mechanics? Some people say the electron is something that has mass and charge, and we can talk about its wave function. Moreover, every "atoms" have natural number of electron. I guess I cannot understand what is electron.
 A: One has to keep firmly in mind that physics is about experimental observations and mathematical models that not only fit current observations but also have accurate predictive power for future setups.
Before the end of the 19th century "particles" meant  objects characterized with a mass that followed classical mechanics ( Newtonian physics), and  radiation which followed Maxwell's equations. It is instructive to read the histories of how  atoms and electrons and photons were first observed. The electron in particular has a long experimental story.

In 1896, the British physicist J. J. Thomson, with his colleagues John S. Townsend and H. A. Wilson,[13] performed experiments indicating that cathode rays really were unique particles, rather than waves, atoms or molecules as was believed earlier.2 Thomson made good estimates of both the charge e and the mass m, finding that cathode ray particles, which he called "corpuscles," had perhaps one thousandth of the mass of the least massive ion known: hydrogen

Since then a "particle" tag has been following the electron , up to the standard model physics table of elementary particles.
In a simplistic manner an electron is a particle that hits the cathode ray tube television surface and gives the images . It has mass, it hits a point at (x,y) on the screen and fits the form of a classical particle.
Nevertheless it is not confined to classical particle characteristics. At the microscopic scale it obeys the laws of quantum mechanics, not classical mechanics. This means that in specific experiments, as the double slit experiment,  even though one electron at a time is used, an interference pattern may appear in the accumulation of hits on a screen, indicative of the solutions of the quantum mechanical wave equation, which predicts the outcome of such experiments as probability distributions. The last link gives the "axiomatic" assumptions of quantum mechanics, which successfully predict experimental results. It is postulate number 1 that assumes particles exist ( as do newtonian physics)
Using the Schrodinger equation and an electron in the electric field of a proton one gets the Hydrogen atom wave function, and the solutions for transitions from the energy levels fit the spectra measured. This validates the quantum mechanical postulates and again the electron's  behavior has sine and cosine shapes indicating a wave nature for the probability of finding one around hydrogen. A recent experiment has recorded this.
So an electron is a particle having the attributes given in the table of particles of the standard model and its behavior is governed by quantum mechanical equations which give probabilities for measurements of position and energy, and not classical trajectories.
A: Today, nobody really knows what essentially an Electron is.  Nobody knows what it is really made of.  We know some of its properties which we can use to manipulate it and exploit it for our needs.  Today, the best we can say concisely about the Electron is that it’s a statistical cloud of an unknown substance.  If you want to go deeper, please read below.
In the era of Classical Physics (before the discovery of the wave-particle duality of matter’s basic constituents), the Electron was considered to be one of the fundamental and indivisible constituents of the matter.  Also then, nobody could say what it is really made of.  But the scientists could at least relax in the notion that we can define with infinite precision the Electron as a rigid sphere of mass which obeys consistently and with infinite precision all the well-known precise laws of the Classical Physics.  But the discovery of the wave-particle duality of Electron, followed by the duality’s mathematical framework known as the Quantum Mechanics, proved that the above notion is just an idealized concept based only upon our macroscopic experience of the reality.  Since the 1930s, the Electron and other fundamental particles are considered equally as both waves and particles.  They reveal either one of those aspects to us depending upon the way we trace or measure them.  There is no objective precedence to either one of the above aspects (wave or particle).  The chances to find an Electron in a certain location are only statistical and cannot be calculated precisely with the Classical Physics laws.  
However, later theories try to give more specific answers to the wave-particle duality.  There’s the Bohmian theory by David Bohm which takes the electron authentically as a real classic particle which is just accompanied and influenced by some abstract waveform.  This view could be supported by the relatively recent experimental, but accidental, discovery by Yves Couder and colleagues made by using simple oil droplets.  A very good review of this experiment and its possible meanings is found here: 
https://www.quantamagazine.org/20140624-fluid-tests-hint-at-concrete-quantum-reality/
Another promising theory is the Transactional Interpretation (TI) of Quantum Mechanics as first proposed by John G. Cramer in 1986.  It was recently expanded more by Ruth E. Kastner who has published a whole book about it in 2012.  A good review about this book is found here:
http://ieet.org/index.php/IEET/more/scaruffi20150108
In contrast to the above Bohmian Theory, The TI theory at its bottom line says that the authentic nature of Electron and other quantas is the waveform.  Furthermore, all the quantas, including the Electron, in their original waveforms, exist outside of our known space-time.  She calls that reality ‘Unactualized Physical Reality’.  She argues that the particles as we know them are the actualized forms of those quantas which transact with each other in the Unactualized Reality and these transactions give rise to the materialistic reality as we see it.  She continues and argues that the whole actualized reality of matter and space-time as we experience it, continues to exist, because those transactions between the actualized-quantas (particles) keep happening ceaselessly in our space-time.  
The origins of the TI theory are in some abnormal solutions of the Maxwell Equations of electromagnetism.
A: There was a lot of temptation and attempts to present the electron as a wave or as a wave packet. But the problem arose, because such waves and wave packets are quickly destroyed because of dispersion.
But there is a case where the dispersion disappears. This is the case when the group velocity of wave propagation becomes equal to the phase velocity.
However, the equations of quantum mechanics that existed up to now did not allow one to obtain such a wave without dispersion. Because the resulting dispersion relations did not have a point of intersection of the phase and group velocities.
Pay attention to the interesting work where this problem is solved.
http://vixra.org/abs/1710.0239
