In the “Standard Model” we have a very successful mathematical model of the known particles of matter and of the strong, weak and electromagnetic forces that act on them. Take a look at the Review of Particle Physics. You will find page upon page of tabulations of properties of elementary particles and their reactions, many measured with remarkable precision. And all of it is explained by the Standard Model!
But the thing is, absent a Higgs field the Standard model says the fundamental particles, like quarks and leptons, as well as the mediators of the weak interactions, are massless. But they ain't. And if there is a Higgs field, there must be a higgs particle: just like quantum fluctuations of, say, the electromagnetic field manifest themselves as particles we call photons, so do quantum fluctuations of the higgs field manifest themselves as particles, “higgs bosons.” Sure, there are alternative theories explaining the masses of fundamental particles that don't contain a particle like the higgs boson, but (1) they are ugly compared to the Standard Model, and, most importantly, (2) the higgs boson has been seen! Check out the higgs-observation pages of the ATLAS and CMS experiments at CERN.
My higgs-boson-related research has addressed several questions. The first one is clear: how do we know what we have seen is the higgs boson? If it walks like a duck and quacks like a duck, … may it be a scaup or a smew? It turns out that there is a large class of models that contain no higgs field, but they all predict the existence of a particle called a “dilaton” and, my collaborators and I showed, in first approximation the dilaton behaves just like a higgs! This begs the next question, how do you tell them apart? Or more generally, how do you know it is the higgs of the Standard Model? The answer is that you make very careful measurements and compare with theoretical predictions. For example, my students and I made some computations showing that the very rare 3-body decays of the higgs can be sensitive probes of deviations from the Standard Model.
The picture you clicked on, reproduced here, is from a paper with my student Patipan, in which we looked at a simple but popular extension of the Standard Model in which there are two higgs fields. This turns out to mean there are two neutral higgs particles, a charged higgs particle (plus its oppositely charged anti-particle) and a third neutral particle, a kind of cousin of the higgs. Hypothesis: one of the two neutral higgs particles corresponds to the particle observed at CERN. Question, from the measured properties of that particle, what can you say about the others. The figure has one of the answers. It gives, as a function of the unknown mass of the second neutral higgs, what fraction of the time that second higgs decays into different combinations of particles. With other students we later extended the study to include whole classes of extensions of the Standard Model and found theoretical upper bounds on the masses of the hypothetical particles (good news for LHC experiments that, after all, are limited in the range of masses of particles they can produce). (Note: it is fundamental constituents (like quarks and leptons) of the elementary particles (like baryons and mesons) that are mass.