Which particles does the Higgs Field give mass to?

The Higgs field $\phi$ undergoes spontaneous symmetry breaking$^\dagger$ (from a complex doublet to a real scalar field, whose quantum is the Higgs boson) in a process named the Higgs mechanism. $^\dagger$: well it's a local/gauge symmetry, not global, so it's not "real" SSB, hence the different name "Higgs mechanism".

This has two consequences:

• the gauge bosons $W^\pm$ and $Z^0$ acquire a mass term, which they couldn’t have had a priori without breaking gauge invariance. The mass depends on the VEV (vacuum expectation value) of the Higgs field, but it doesn’t arise from a direct interaction term (see below) between the Higgs field and the gauge bosons. This new mass mode is the “would-be” Goldstone boson associated with the breaking of the Higgs field symmetry. (By “no direct interaction term” I mean that the term containing a product between the gauge boson and the Higgs field is hidden in the gauge covariant derivate $D^\mu$.)

• the fundamental fermions (quarks, leptons, but not neutrinos) also acquire a mass term. This arises from a direct interaction term between the fermionic field $\psi$ and the Higgs field, called the Yukawa Lagrangian sector. This looks like $ \mathcal{L}_Y \propto \Gamma \bar L \phi R, $ where $\Gamma$ is the Yukawa coupling to the specific fermionic field $\psi$, and $L$ and $R$ are the left- and right- handed components of $\psi$. Neutrinos have no right-handed partner so they cannot gain mass through a Yukawa coupling.

So the Higgs field is responsible for the masses of all the elementary particles (including the Higgs boson) short of neutrinos.

In the Standard Model, the Higgs field also gives mass to the six quarks (up, down, strange, charm, top, bottom) and the three charged leptons (electron, muon, tau) through Yukawa couplings. Some related mechanism may give neutrinos a small mass. (They’re massless in the Standard Model, but we know this is wrong.) Finally, one can argue that the Higgs field gives the Higgs boson its mass.

In short, every elementary particle except neutrinos gets its mass from the Higgs field. We’re not sure yet how neutrinos get theirs.

The Higgs field is coupled to the fermions (quarks and charged leptons) in the standard model via Yukawa couplings. As a result of the Higgs mechanism, the Higgs field then gives mass to these fermions, in addition to the weak bosons.