The Resonance Signal and Its Significance
The ATLAS experiment at the large hadron collider (LHC) has seen a signal at 3.5 sigma significance of a potential new particle with a mass of about 750 GeV/c^2 in diphoton events. The CMS experiment has also seen a signal, of about 2 sigma significance, in diphoton events of roughly the same mass. The diphoton channel is a particularly clean way to discovery new particles because there is not much background from Standard Model events to interfere at that mass scale. CMS has also seen a 2.5 sigma signal at roughly the same mass in the charged lepton-neutrino-quark-anti-quark pair channel.
The data from the individual experiments standing alone and after considering look elsewhere effects, is not that significant, but the confirmation from two independent experiments makes these moderately significant bumps seem much more significant. While this doesn't amount to the discovery of a new particle (considered to be 5 sigma evidence), it is the most credible evidence yet that there could be a new particle.
I would put the likelihood of this resonance being real at about 45%, and the likelihood of this resonance being both real and unexplainable by Standard Model physics (e.g. for it not being a Standard Model composite particle of some sort) at about 35%.
As a consequence of conservation of intrinsic angular momentum (colloquially called "spin" even though that term has multiple other meanings), a particle that decays in the diphoton channel would have to be either "scalar" or "pseudo-scalar" (i.e. spin-0), or tensor (i.e. spin-2), and would have to have zero electric charge.
In other words, this particle looks a lot like a heavy Higgs boson. It is also hard to reconcile a heavy Higgs boson which would be expected to have a wide width, with the narrow apparent width of the "bumps" that are actually seen (the width is about 40 GeV). There is also no particularly well motivated reason to think that this would be a graviton resonance, even though that could, in principle, produce a spin-2, zero charge diphoton resonance.
Potential Theoretical Explanations
Most prosaically, this could be a case where six ordinary Standard Model Higgs bosons are produced at the same time (e.g. through the fusion of multiple gluons at once) and they are synchronized or superimposed upon each other in some manner that causes their combined decay product to be a diphoton decay. It could be that there are similar bumps at 250 GeV and 500 GeV but that those have not been as apparent because of much larger backgrounds at lower energies.
Supersymmetry (aka SUSY) and a lot of other beyond the Standard Model theories predict the existence of "two Higgs doublets", with a total of five Higgs bosons, a positively charged one, a negatively charged one, an extra "scalar" Higgs boson, and an extra "pseudoscalar" Higgs boson.
Marco Frasca, a physicist whose primary interest is QCD, meanwhile, has argued that the Standard Model Higgs boson should have an infinite number of higher energy excitations.
While lots of models predict such a particle, however, there is less clarity over what kind of couplings this particle would have to other particles (i.e. "what does this particle do?"), and many predict that an extra Higgs boson would be accompanied by other particles (often lighter than the heavy Higgs boson) that have not been discovered.
Importance, If Real
In a lot of ways, a new fundamental particle at 750 GeV would be far more significant than the discovery of the Standard Model Higgs boson.
The Standard Model Higgs boson was predicted to exist forty years before it was discovered and was necessary to the consistency and good functioning of the rest of the Standard Model. When it was discovered, every fundamental particle in the Standard Model had been discovered and none of the particles not predicted by the Standard Model had been discovered.
If there is a new fundamental particle at 750 GeV, however, this is definitive evidence of beyond the Standard Model physics of some kind, although the extensions could be very narrow, for example, in the model that Marco Frasca suggests, or very broad, for example, in the case of non-minimal supersymmetry.
Of course, it is also possible that this signal could be some non-fundamental particle (e.g. an excited state of a spin-0 glueball or an excited state of top quark quarkonia) which would still be very interesting, but far less interesting than a new fundamental particle.
As usual, the next step is watchful waiting as the ATLAS and CMS experimenters do their jobs.
Jester has more analysis. Physics Forum discusses it here. Not Even Wrong coverage here. Matt Strassler is cautious and skeptical. Dorigo discusses the results but has little analysis.
Today is the last day that the LHC was collect data in 2015. It will start operations again and start collecting more data sometime around April in the year 2016.