The Higgs Mechanism is really quite complex and involves a /lot/ of particle theory.

Basics of particles: There are six quarks; up, down, top, bottom, strange, and charm. These can form into mesons (a quark and and anti-quark) or baryons (three quarks). Overall particles made up of quarks are called hadrons. There are also particles like electrons, etc, which aren't made up of quarks. Electrons are leptons. Now, there are also the particles that are force carriers. For electromagnetism, this is the photon. For the weak force, these are the W and Z bosons. For the strong force, it's the gluon. For gravity we have the (theoretical) graviton. We say these particles are the force carriers because, for instance, in electromagnetic interactions, a photon is exchanged. Now, gluons and photons are massless, whereas the W and Z bosons are not. There are reasons that this is true, but they'd take a long time to explain.

Now, particles in general are subdivided into bosons (meaning they have integer spin) or fermions (meaning they have half-integer spin).

Now, the Standard Model (which basically is a theory uniting our understanding of particle physics and three of the four fundamental forces) doesn't exactly give a method by which particles have mass. This is where the Higgs boson comes in. Through spontaneous symmetry breaking (which is really just impossible to explain in layman's terms) we find a certain number of massless Goldstone bosons, and one massive boson (the Higgs). Long story short, this allows us to understand how general particles have mass.

One sentence summary: The Higgs mechanism (which works through the Higgs boson) gives particles mass.

Finding the Higgs is cool because it means we understand why particles have mass, and it means our current understanding of particle physics is fairly correct.