The Higgs boson is a theoretical particle that is part of the Standard Model of quantum physics. It is a short-lived particle that is a physical manifestation of the Higgs mechanism, a theory proposed in 1964 by the British physicist Peter Higgs, who expanded on the ideas of American theoretical physicist Phillip Anderson.
In the Standard Model, space consists of the Higgs field, with a non-zero value in all space. There are two neutral and two charged components to the field. One of the neutral and both of the charged components combine to create the W & Z bosons, which create the weak force, one of the fundamental forces of physics.
The remaining neutral charge creates the scalar Higgs boson, which has neither charge nor spin (thus causing it to follow Bose-Einstein statistics, and making it a boson). This is crucial in using the Standard Model to explain where the mass of the W & Z bosons come from.
Important Note: The Higgs boson isn't needed to explain all of the mass. Most of the mass in our bodies, or the rest of the universe, comes from the strong nuclear force, which bonds quarks together to form other particles, and also binds those particles together in turn. It's this process that holds together the protons and neutrons in an atomic nucleus, for example, despite the electromagnetic repulsion among protons. The majority of mass can thus be fully explained without resorting to the Higgs mechanism.
The real reason physicists care so much about the Higgs boson is that it is crucial for understanding the spontaneous broken symmetry in the electroweak force. (Again, it's this broken symmetry that gives the W & Z bosons their mass.)
In May 2010, evidence came to light at Fermilab which suggested there may be as many as 5 types of different Higgs bosons. At this time, physicists are still trying to figure out the implication of these results. If the Large Hadron Collider is able to successfully create Higgs bosons, then it will be possible to test the theoretical predictions in greater details to understand more about the particle.
The Higgs boson is the only Standard Model particle has not been observed experimentally, though recent evidence, announced July 4, 2012, indicate that it might have manifested within the Large Hadron Collider. Further evidence needs to be collected to determine this for sure, although at present the Boson discovered at the LHC looks like it may well be the most conventional type of Higgs boson, rather than a particle that hints at more exotic physics, such as supersymmetry.