The property of mass that almost every particle possesses comes from the Higgs Field. It is this field, which permeates all of space, that particles interact with and hence obtain mass.
But why do some particles have a higher mass if they are all interacting with the same Higgs Field? One would suspect that they'd all have the same mass, but there must be some property of a particular particle that makes it interact with the field in a different way, thereby giving it mass.