the Higgs mechanism
Here is the central trick that rescued particle physics from the mass problem. The theory forbids you from simply writing a mass into the equations for the W and Z bosons; doing so wrecks the symmetry and ruins the predictions. The Higgs mechanism is the clever workaround: instead of inserting masses by hand, you add a new field (the Higgs field) whose lowest-energy state is nonzero, and let the force carriers acquire their mass by interacting with that ever-present field. The symmetry of the equations is kept exactly intact; it is merely hidden by the state the world settled into. Mass appears as a consequence, not as an assumption.
Concretely, the Higgs field's nonzero vacuum value reaches into the equations for the W and Z bosons and effectively gives them a mass term that the symmetry would otherwise have banned. There is also a striking bookkeeping detail. A massless force carrier like the photon has two possible polarizations (two ways of wiggling), but a massive one needs three. The extra, third polarization does not come from nowhere: it is supplied by one component of the Higgs field, the would-be massless Goldstone boson, which is absorbed into the W and Z. Physicists say the gauge boson 'eats' the Goldstone boson and thereby gains both mass and its third polarization. The same mechanism, applied to the matter particles via Yukawa couplings, gives the electron and quarks their masses too.
The Higgs mechanism is the keystone of the Standard Model: without it the unified electroweak theory would predict massless W and Z bosons and a world utterly unlike our own. It was proposed in 1964 and earned Englert and Higgs the 2013 Nobel Prize, the year after the field was confirmed. The honest caveat, worth repeating: the mechanism explains the masses of fundamental particles, not the masses of composite bodies. Most of an atom's mass lives in its protons and neutrons, whose mass is overwhelmingly the energy of the strong force, so the Higgs accounts for only a small slice of the mass of ordinary stuff.
Before symmetry breaking, the electroweak theory has four massless force carriers and a four-part Higgs field. After breaking, three of the Higgs parts are 'eaten' to make the W+, W-, and Z heavy, one part survives as the photon staying massless, and the fourth Higgs part is left over as the Higgs boson we found in 2012.
Three Higgs components become the heavy W/Z's third polarization; one becomes the Higgs boson.
The Higgs mechanism gives mass to fundamental particles, not to composite ones; saying 'the Higgs gives everything its mass' is wrong, since most everyday mass is strong-force binding energy.