The "vacuum" is just the ground state of a QFT. Here we discuss the Standard Model, and within the Standard Model the ground state is such that the Higgs field's expectation value in the vacuum state is non-zero. Due to Poincare invariance of the vacuum thie vacuum expectation value (vev) must be homogeneous and isotropic. Indeed within the standard model it provides all the fundamental masses to the matter particles through the coupling of the corresponding fields to the Higgs field (i.e., the lepton masses and the current-quark masses) in the Standard Model (let's not consider neutrino masses at the moment, which are already kind of "physics beyond the Standard Model").
Note, however, that only about 2% of the mass of the matter around us is due to the Higgs mechanism. The remaining 98% of the mass is dynamically generated through the strong interaction, binding together quarks and gluons to colorless objects ("color confinement"). This is, however, a highly non-perturbative phenomenon and thus not really understood today. We, however, are pretty sure that it's the correct idea from lattice-QCD calculations which give a pretty accurate account of the mass spectrum of the known hadrons (mesons and baryons, among the latter protons and neutrons as the fundamental building blocks of the atomic nuclei building up the matter around us).