As a non-physicist, my understanding of the Higgs mechanism is pretty weak, but...
The way I understand it the generic idea of the Higgs mechanism isn't simply restricted to be a "mass-generating" mechanism, although the most interesting Higgs mechanism that are searched for are the ones that can suggest electro-weak symmetry breaking which gives non-zero rest masses to W and Z bosons. The reason why most folks are searching for something that can describe this is that W and Z boson have the largest masses (~85x a proton) whereas a simplistic assumption of symmetry in the electric and weak fields would imply that these particles have zero-mass.
The generic Higgs mechanims pre-supposes a field (which exists everywhere there is the effect). Simplistic assumptions of convervation properties of the field manifest themselves as certain symmetries (which imply other symmetries and can be put into a mathematical framework similar to gauge groups, but that's not too important). The question becomes since the classical equations describe a mass-like scalar term that seems to be independent of the gradients and couples to these fields, how can we put a mass-like term into a framework with all these symmetries, yet be have it be non-zero?
The basic idea is that the field must have some sort of symmetry-breaking. At some high-energy (or temperature or potential), the expected value of the mass-like term averages out to zero, but as you go towards low-energy (or temperature, or potential), the expected value becomes non-zero yielding a mass-like term. As it turns out, if the potential of the field is shaped like a W or a sombrero, it can have this property. At high potential (when you can't see the "hat" part), the expection is symmetric around zero and the field has the required symmetry. As the potential goes down (including all the way to zero), the low-energy expectation value falls away from zero (at a consistent radius away from the center because of the "hat"). This is apparently called spontaneous symmetry breaking.
Why would the field potential have this shape? Who knows? But if it does, it can simultaeously satisfy the symmetries and still have a gradient/direction independent expected value term (which would act just like a scalar constant just like mass in all the classical physics formulas).
So what everyone is calling the Higgs field (and higgs particle) are mostly a specific interaction that describes how the W and Z boson can appear to have a non-zero mass-like term with electro-weak symmetry breaking. It is my understanding that it is currently just assumed that all other stuff that has non-zero mass would be similar, but it doesn't really explain what people think of as mass per-se (which has a mysterious gravitational equivalence), it just gives a explanation for the observations that are likely consistent with mass in terms of energy of the various fields (other than gravity).