There is a single Higgs field. But the Higgs field can be excited in many different places! Same for anything else, like an electron - that's why we say that two electrons are indistinguishable from each other. Fundamentally, they're made of the same "stuff": the electron field that permeates the universe.

Particles get mass by "coupling" to the Higgs field. I.e. the excitations of the electron fields - electrons - interact with the excitations of the Higgs field - Higgs bosons. This, through a complicated process described by quantum field theory (that I don't admit to fully understand), gives the particle what we call mass. Different particles couple to the Higgs with different strengths, yielding various particle masses. The Higgs even couples to itself (rather incestuously), which is why it has mass!