For example, when we meassure Higgs boson mass to be 125 GeV, do we think about renormalized or pole mass? Should the mass of the Higgs change if it is produced at higher energies?
The pole mass is closer to the intuitive physical mass of a particle, and is typically what is reported by experimentalists. The jargon comes from the well-known fact that resonances (and stable particles) show up as simple poles in the scattering amplitude continued to complex kinematic variables. This mass does not change with energy.
The 'running mass' refers to a parameter in the Lagrangian, of mass dimension = 1. This parameter is to be treated as if it were just another coupling constant; and just like any coupling constant in QFT, it changes with (the renormalization) scale.
A calculation can be done to relate the two, typically done perturbatively by computing the self energy. Thus, when an experimentalist quotes a mass (which is almost always the pole mass), this fixes the value of the running mass (in some renormalization scheme) by the said relation.