The standard model of particle physics has one major problem. The particles are all massless. This uncomfortable situation was remedied using a clever trick by Peter Higgs way back in 1964. The mathematical details can be found in Quantum Field Theory Demystified. We won't review that here, we'll just note that Higgs postulated the existence of a field, now aptly named the Higgs field, which fills all of space-time. Particles interact with this ever present vacuum field, and just like anything else, different particles interact with different strengths. The strength of the interaction determines the mass of the particle. If there were no Higgs field all particles would travel at the speed of light.
I saw a talk once, I don't remember who gave it, where the guy likened the Higgs field to water in a swimming pool. Imagine being underwater and moving your arm up and down. So by analogy you can kind of think of the resistance of the water to the interaction of a particle with the Higgs field. Like all fields, the Higgs field has a particle associated with it. Its called the Higgs boson, and would be the only known fundamental particle with zero spin.
One of the first items of business for the Large Hadron Collider (LHC) is going to be finding the Higgs. But what if they don't find it? Surprisingly there are some other ideas that explore the acquisition of mass by particles in the standard model. So either way, whatever comes of the search for the Higgs at the LHC is going to lead to some interesting physics down the road.
Here is a link to a paper exploring one of these ideas, for the more mathematically inclined.
Soluble Theory of a noncompact Group