The Higgs boson is a hypothesised particle which, if it exists, would give the mechanism by which particles acquire mass.
Matter is made of molecules; molecules of atoms; atoms of a cloud of electrons about one-hundred-millionth of a centimetre and a nucleus about one-hundred-thousandth the size of the electron cloud. The nucleus is made of protons and neutrons. Each proton (or neutron) has about two thousand times the mass of an electron. We know a good deal about why the nucleus is so small. We do not know, however, how the particles get their masses. Why are the masses what they are? Why are the ratios of masses what they are? We can't be said to understand the constituents of matter if we don't have a satisfactory answer to this question.
Peter Higgs has a model in which particle masses arise in a beautiful, but complex, progression. He starts with a particle that has only mass, and no other characteristics, such as charge, that distinguish particles from empty space. We can call his particle H. H interacts with other particles; for example if H is near an electron, there is a force between the two. H is of a class of particles called "bosons". We first attempt a more precise, but non-mathematical statement of the point of the model; then we give explanatory pictures.
In the mathematics of quantum mechanics describing creation and annihilation of elementary particles, as observed at accelerators, particles at particular points arise from "fields" spread over space and time. Higgs found that parameters in the equations for the field associated with the particle H can be chosen in such a way that the lowest energy state of that field (empty space) is one with the field not zero. It is surprising that the field is not zero in empty space, but the result, not an obvious one, is: all particles that can interact with H gain mass from the interaction.
Thus mathematics links the existence of H to a contribution to the mass of all particles with which H interacts. A picture that corresponds to the mathematics is of the lowest energy state, "empty" space, having a crown of H particles with no energy of their own. Other particles get their masses by interacting with this collection of zero-energy H particles. The mass (or inertia or resistance to change in motion) of a particle comes from its being "grabbed at" by Higgs particles when we try and move it.
If particles no get their masses from interacting with the empty space Higgs field, then the Higgs particle must exist; but we can't be certain without finding the Higgs. We have other hints about the Higgs; for example, if it exists, it plays a role in "unifying" different forces. However, we believe that nature could contrive to get the results that would flow from the Higgs in other ways. In fact, proving the Higgs particle does not exist would be scientifically every bit as valuable as proving it does.
These questions, the mechanisms by which particles get their masses, and the relationship amongs different forces of nature, are major ones and so basic to having an understanding of the constituents of matter and the forces among them, that it is hard to see how we can make significant progress in our understanding of the stuff of which the earth is made without answering them.