What sort of "mass" is explained by the Higgs mechanism? When I asked this question (probably in a less neutral form) to physicists, their answer was something along the lines that it's not gravity (i.e. unrelated to gravitons) but inertial mass. (So I wondered whether this is an analogous mechanism to gravitons, only that it explains inertia.) Now after some weeks of thinking (and reading) about this, I think I finally figured out what they were trying to tell me. This is related to the following comment for a similar question:

Have you made up your mind on what "mass" of a particle means to you in that question? Maybe that will help.

For me, the obvious candidates what "mass" might mean are


*

*gravitational mass

*inertial mass

*rest mass


My current guess is that the Higgs mechanism explains why "other particles" (only fermions, or also other bosons?) have a non-zero rest mass. (I imagine it's some form of explanation for potential energy related to the mere presence of the particle, even in the absence of "interactions" with other particles.) However, at least some of the (popular science) explanations really seem to try to explain something related to motion and inertia, and I got the answer "inertial mass" so often that I wonder whether it's actually really the "inertial mass" (of fermions) that is "directly explained" by the Higgs mechanism (this doesn't preclude that this explanation might be "translated" into something equivalent to rest mass).
 A: General relativity doesn't care about the difference between mass and energy. In the stress-energy tensor T$_{00}$ is the energy density and mass is just treated as energy divided by $c^2$. GR doesn't care what the Higgs' mechanism does, and will work just as well above the electroweak transition where the particles (well, the vector bosons at least) are all massless.
What the Higgs' mechanism does is explain why everything doesn't travel at the speed of light. It's the constant of proportionality between velocity and momentum. I guess in your terms it's the inertial mass/rest mass.
A: Think of the Higgs mechanism as affecting rest-mass.  This is the mass that a particle has when it is sitting still (you can weigh it to figure it out).
Think of gravity as affecting energy.  More energy = more gravitational force.  So an electron that is moving very quickly has a total energy of its rest mass (E = mc^2) + its kinetic energy.
Consider an electron and a positron.  These both have rest-mass.  When they collide and turn into two photons, all rest-mass is gone.  Energy is conserved, so the system still weighs the same at all times.  But the Higgs mechanism only affects the electron and positron, not the photons.
