What is "mass" in particle physics? It's clear, from reading pop-science articles about the Higgs boson, that particle physicists have something very specific in mind when they say "mass". In classical physics the mass of a particle is just a given value, but in the context of particle physics you hear things like "computing the mass" or "such and such interaction gives some mass".
What does "mass" mean to a particle physicist? How does it relate-to/explain the classical notion of mass, and how does it differ?
I'm not sure what level of explanation I'm looking for. I don't know particle physics, obviously, so any grad-level equations are likely to go way over my head. But explanations like "bouncing back and forth all the time" from this minute physics video are too simplified. I'm left wondering, for example, how coupling with the Higgs field makes electrons bounce back and forth? And why wouldn't it just scatter them? And how is this process Lorentz invariant? And why does the bouncing resist/react-to forces?
 A: Particle physics has a set of elementary particles , some of which have zero mass.

In this table the general public has heard of the electron and maybe the photon. The mass of the electron has been measured consistent with classical definition of mass.
The mathematical theory of elementary particles is called the standard model and the table has the masses that enter in the lagrangian.  These are the only masses that are generated by the Higgs mechanism.
Due to the fact that special relativity reigns at the particle domain, the invariant mass defined as this mass.

In the case of the particles in the table this mass and the mass in the table coincide. When though there are combinations of particles, or composite ones, as the proton and the neutron which are composed of quarks and gluons, the mass of the composite is the invariant mass, i.e. the energy momentum vectors of all components added up in the invariant mass formula define the mass of the composite. It is evident that in  this case the intrinsic mass generated by the Higgs mechanism has a small part to play. The mass of the proton is also measured by classical methods

I'm left wondering, for example, how coupling with the Higgs field makes electrons bounce back and forth? And why wouldn't it just scatter them? And how is this process Lorentz invariant? And why does the bouncing resist/react-to forces?

This is really another question, on how the Higgs mechanism works. Have a look here and here . The popularized explanations can only be approximations.
A: The mass always means the same thing – but in different theories, one uses different equations and other tools to express the mass.
Inertial mass $m$ is the quantity expressing "resistance of the object with respect to acceleration", i.e. the coefficient that enters Newton's $F=ma$. The gravitational mass is what enters the formula for the gravitational force $F=GmM/r^2$. These two masses are the same, up to a universal ratio that may be chosen to be one in good units. This equality of the two masses is known as the "principle of equivalence".
In Newtonian physics, the mass of individual particles is basically conserved. Surely the total one is conserved. In special relativity, only the "total relativistic mass" which includes the enhancement from the kinetic energy is conserved. This conservation law becomes the same law with the energy conservation law; the mass may be "converted" to energy and vice versa. The total mass and the energy are related by $E=mc^2$ and this "unified" quantity is conserved whenever the laws of physics are time-translationally invariant (Noether's theorem).
For a given particle, in special relativity, we usually use the term "mass" for the rest mass $m_0$, the mass measured in the rest frame. It's related to the energy and momentum by
$$ E^2 = m_0^2 c^4 + p^2 c^2 $$
The actual working theory used to describe practical particle physics is called quantum field theory (QFT). It's a combination of classical field theory (like Maxwell's electromagnetic fields) and the postulates of quantum mechanics.
In QFT, particles are created and annihilated by fields, they are quanta of these fields. Each particle has a value of the mass and it's encoded in the "mass term". For example, for a scalar boson, the Lagrangian is
$$ L = \frac{1}{2} \partial^\mu \Phi \partial_\mu \Phi - \frac {m_0^2}{2} \Phi^2 $$
Why is it the same mass as I defined at the beginning? It's because the field equations resulting from the Lagrangian above are the Klein-Gordon equation
$$(-\square - m_0^2 )\Phi =0$$
The so-called plane waves
$$ \Phi = A\exp (ip_\mu x^\mu) $$
are the simplest solutions to the Klein-Gordon equation. The momentum must obey $p_\mu p^\mu =m_0^2$. In the rest frame, $\Phi$ is independent of space  but it depends on time via
$$\Phi = A\exp(-i m_0 c^2 t / \hbar) $$
where I restored the $c$ and $\hbar$ i.e. the usual units. But the angular frequency of this associated wave $\Phi$ is related to the energy via $E=\hbar\omega$, a basic formula of the quantum theory. 
So the mass $m_0$ is measured as $E/c^2$ in the rest frame. The energy is calculated from the angular frequency $E=\hbar\omega$. And the angular frequency may be adjusted by adjusting the mass term $-m_0^2/2 \Phi^2$ in the Lagrangian. That's why this mass term controls the usual inertial mass.
For fermions, the box is replaced by the Dirac operator and the second power of $m_0$ becomes the first power. Different particle species have different values of the mass. Anna's answer is conveniently complementary and addresses this part of the question.
At any rate, the mass is in no way "redefined" in particle physics. It exactly agrees with everything we expected from the "mass" in the past. The equations needed to describe the creation and annihilation of elementary particles are given by QFT and QFT has to change the value of the masses – and it's still the same concept of "masses" – through the "mass terms".
