So on Special Relativity: What is mass, according to physics?

Mass, uh? This is an interesting topic.

The classical definition of mass is essentially how much matter is contained in a body. Like how much stuff is in you. It also determines the strength of the gravitational force you exert over other materials and they exert over you.

In the past, you three kinds of masses. The inertial mass, that represents the resistance of a body to an applied force. You can think of it this way, for the same force, a body that has more mass, would move less, because it would suffer a lower acceleration. That is all condensed in Newton’s second law of motion: F⃗ =mia⃗   
$\overrightarrow{F}=m_i\overrightarrow{a}$.

The other kinds of mass are called gravitational masses. Active gravitational mass of a body is the strength of gravitational force exerted by that body, or F⃗ g=Gmr2Mg${\overrightarrow{F}}_g=\frac{Gm}{r^2}{M}_g$. And Passive gravitational mass is the gravitational force exerted on that body in a gravitational field or Fg→=mgg⃗ .$\overrightarrow{F_g}=m_g\overrightarrow{g}.$

For quite a long time, physicists were trying to figure out whether inertial mass and the gravitational masses were the same. Einstein proved that there was no real difference between inertial mass and gravitational mass, so there was no real experiment that could differentiate the two. To think of it in a different way, assume you were stuck in a box, and you had no way of actually knowing what was outside that box. Now assume you were in a place in space, too distant to suffer significant gravitational force from anything. If we put rocket boosters in your box, with acceleration = 9.8 meters per second (about the same as the Earth’s gravitational acceleration), would you be able to notice the difference? Einstein concluded that you couldn’t, and that principle is called the equivalence principle, being the basis for General relativity. Also Einstein showed a equivalence in mass and energy, in Special Relativity (specifically Rest mass and Rest energy), because the true expression for a moving body is E2=m2c4+p2c2$E^2=m^2{c}^4+p^2{c}^2$, with p being the momentum of the body. If the body is not moving, p is equal to zero, and you get the famous E=mc2($E=m{c}^2$. Now this other equivalence is very important, because, considering that inertial mass is equivalent to gravitational mass, and inertial mass is equivalent to energy, that means that bodies without mass can also suffer gravitational force (i.e. photons).

As for where does mass come from, the modern quantum field theory says that is comes from the interaction of quantum fields of different particles with the Higgs field. You see, for the modern quantum field theory, all of the particles can be described as actually being oscillations in quantum fields, which are massless in their nature. In order for them to gain mass there needs to be a coupling of the respectively quantum fields with a Higgs field, breaking the symmetry of the system and getting this mass, as far as I understand the subject, which is not a lot. But we should definitely go back to this subject when we finally reach QFT.