Particle physics up to now has come up with the Standard Model as a mathematical theory that fitted most experimental data up to the LHC and was successful in predicting the results of experiments, including many analysis of LHC data. The standard model has one Higgs, that became necessary because just the groups structure that the data implied ( SU(3)xSU(2)xU(1) ) was not sufficient to go on to predictions since all elementary particles were presumed massless. The Higgs field, as explained in the video you linked to, was the tool to give the measured masses to the observed particles. The existence of the Higgs field in the standard model directly implied that a Higgs boson should exist. Thus the 125 GeV boson detected at the LHC has been accepted as the standard model Higgs.
There are reasons to believe that the symmetries of the standard model are not the end of the story, experimental and theoretical. Experimentally we have some inconsistencies as the CP violation problem coming from cosmology, and there is not enough CP violation in the standard model. Theoretically because of ad hoc cut offs in the perturbative expansions when calculating with the SM in order to get finite results and not infinities. Thus one speaks of physics beyond the standard model.
Supersymmetry, which introduces an extra symmetry between fermions and bosons, solves the problem of infinities in calculations. It has more higgses because of the new group structures, depending on the specific supersymmetric model dictated by the new symmetries.
Thus one may say that the 125 Higgs is THE Higgs of the Standard Model, and as people at LHC are searching for supersymetric particles, they are also searching for extra Higgs bosons.
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A point to keep in mind when talking of symmetries in elementary particle physics and the models that describe them is that many of these symmetries are broken as far as the masses of the particles are concerned. The symmetries by themselves give zero masses to everything, it is the broken symmetries with the Higgs mechanism that separate individual particles with the masses we have measured them. The possible existence of many Higgs bosons in higher theories will have them with a range of masses . In this case the 125GeV is the lowest Higgs boson in, for example, a supersymmetric particle spectrum, and will be identified with the expected standard model Higgs. The standard model is not invalidated when it is included/embedded in a higher dimensional model.
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The level of certainty that it is a Higgs is high within the measurement errors.
Whether or not it is a Higgs boson is demonstrated by how it interacts with other particles, and its quantum properties. For example, a Higgs boson is postulated to have no spin, and in the Standard Model its parity – a measure of how its mirror image behaves – should be positive. CMS and ATLAS have compared a number of options for the spin-parity of this particle, and these all prefer no spin and positive parity. This, coupled with the measured interactions of the new particle with other particles, strongly indicates that it is a Higgs boson.
To nail it down as "the" standard model Higgs more data are needed, and I think it will not be nailed down as there probably exists one more Higgs at low energy, pointing to beyond the standard model. We have to wait for the new data. Within errors it looks like "the" Higgs, it is a boson, within errors the branching ratios agree etc. Of course particle physicists hope it is not :), they want new physics beyond the standard model, after all.