In talk shows and fashion magazines the Higgs field is being described as aether filling the entire space, while the Higgs boson is a wave propagating in that medium. This is a useful visualization, but to answer the question from the title we need a better one. A more precise definition is that a Higgs particle is a spin-0 boson whose coupling to other particles of the Standard Model is proportional to their masses. In fact, Higgs is not just another particle than no one ordered; its existence is crucial for the consistency of the theory. The reason is that gauge bosons and chiral fermions are massless in their natural habitats (like photons and gluons), and giving them mass is a serious intervention referred to as local symmetry breaking. Without the Higgs boson (or something else playing the analogous role) a theory with a broken local symmetry loses its predictive power at high energies.
Incidentally, the way the Higgs boson is now being discovered at the LHC has little to do with that important role. In a high-energy hadron collider Higgs is dominantly produced by a fusion of 2 gluons, and is most easily observed via its decays into 2 photons. The couplings to photons and gluons are not really what defines the Higgs, but rather a byproduct induced by quantum corrections. In fact, other scalar particles could enjoy the same couplings without having anything to do with the mass generation (radion is one example). Thus, observing a resonance in the diphoton channel provides a circumstantial evidence in favor of the Higgs boson (given the rate is close to what the Standard Model predicts), but does not directly prove the higgsy nature of the new particle.
Fortunately, we can test whether or not we're dealing with a Higgs. Already now the LHC has a potential to probe the Higgs coupling to W and Z bosons, which are directly related to the mechanism of mass generation. In particular, the LHC is sensitive to the h → ZZ* → 4 leptons decay, where the invariant mass of the 4 leptons should reproduce, to a ~GeV accuracy, the mass of the Higgs. In the 2011 data ATLAS saw 3 events near 124 GeV, while CMS saw 2 events near 126 GeV. It is likely that these events indeed originated from a decay of a 125 GeV Higgs boson, however at that point it could have easily been a statistical fluctuation of the background. On the other hand, the 2011 searches for the h → WW* → 2 leptons + 2 neutrinos decay showed some deficit of events relative to the hypothesis of a 125 GeV Standard Model Higgs. So the jury is still out. Adding the 2012 data should more than double the statistical power: on the 4th of July CMS will update both channels mentioned above, while ATLAS will update the 4-lepton channel only. If they report a signal, that will be a strong evidence that the observed resonance is a Higgs particle. And that is only the first test. In the near future, the LHC will be sensitive the Higgs decays to b-quarks and tau-leptons, as well as production processes via the Higgs couplings to W and Z bosons. Studying the rate of all these processes we can determine whether the new particle indeed couples to mass. Then we'll know whether it's the Higgs, or just one Higgs from a larger Higgs sector, or an impostor.