The various elementary particles, both the matter (quarks and leptons) and force-carrying field particles (photons, gluons, gravitons, and the W and Z bosons) all have a wide range of masses. The photon and the gluon, for instance, have no mass at all, while the W and Z bosons carrying the weak nuclear force are the mass of a good-sized atomic nucleus. Why there should be such a large variation of masses is a major problem for the Standard Model of particle physics, and to solve the problem, Peter Higgs proposed that there should be a field that pervades the whole universe that particles interact with to acquire their masses: the strength of a particle's interaction with this field determines its mass. Like all quantum fields it should have a field boson associated with it, with spin zero as it is supposed to be a scalar field: the Higgs boson.
A useful analogy of this mechanism of the Higgs field giving particles their intrinsic masses would be to imagine a room full of members of a political party quietly chattering (space filled only with the Higgs field). An important politician (a massive particle) enters the room, creating a disturbance as he moves around, attracting clusters of admirers and petitioners alike. The commotion he creates produces resistance to his movement, in other words he acquires mass, just as a particle moving through the Higgs field would. Now imagine that a juicy rumor is whispered into the room. The rumor, as it spreads among the Party members, produces clusters as though someone important were there. These clusters are the Higgs bosons.
The Higgs boson is a vital component of Grand Unified Theories, that attempt to explain why the universe came into being the way it did, why the universe prefers to be filled with matter instead of antimatter, why there is something in the universe rather than nothing. Of course, this whole notion of "the God Particle" as it has been called, is just a theory, and there is, as of this writing, no direct experimental evidence for this field and the particle that is supposed to carry it. It is indeed possible, of course, as Dirac once noted (but speaking of magnetic monopoles) that "pretty mathematics by itself is not an adequate reason for nature to have made use of a theory." There might be other ways for nature to be choosing the masses of particles. However, current theoretical estimates (not from the Standard Model, which by itself cannot predict it, but from more ambitious theories such as supersymmetry and string theory) give the Higgs boson's own mass as somewhere between 60 GeV/c2 (about the mass of an iron atom) and some 150 GeV/c2 (roughly the mass of an atom of the rare earth metal samarium), meaning that only a very high energy event can allow us to observe one, if it does exist (it could even be as high as 1 TeV/c2, but that's something that experimental physicists don't like to think about). The search for this particle is one of the main reasons why ever more powerful particle accelerators such as those at CERN and Fermilab are being built. Finding it would go a long way towards validating the correctness of the Standard Model.
Sources:
Arthur Beiser, Concepts of Modern Physics
The Waldegrave Higgs Challenge at http://hepwww.ph.qmw.ac.uk/epp/higgs.html
http://www.aip.org/enews/physnews/1996/split/pnu298-1.htm
http://lutece.fnal.gov/Drafts/Higgs.html