The Cosmic Microwave Background (CMB) is radiation we receive today from a time when the universe was about 300,000 years young. At that time, radiation decoupled from matter and since then, photons could travel almost undisturbed. The CMB shows the temperature, or the inverse wavelength, of the microwaves that we receive on Earth from these early times.
The mean temperature of the CMB is approximately 2.7 Kelvin, and is a blackbody spectrum to truly amazing accuracy. What we will be concerned with here however is not the mean temperature, but tiny fluctuations around this temperature. These carry a lot of information about the conditions in the early universe which can help us understand the origin of the structures that we see today, and the processes that were important in the early universe. These fluctuations are of the order micro-Kelvin, and have been measured by NASA's WMAP mission. You probably have all seen their skymap of the temperature fluctuations:
We have discussed the features and usefulness of the CMB temperature fluctuations a few times already, see e.g. my earlier posts The CMB Power Spectrum and Anomalous Alignments in the CMB.
One way to extract information from the data is to look at correlation functions. These come in integer orders like the two-point function, the three-point function, the four-point function etc. There also is a one-point function but ‒ assuming a homogeneous probability distribution ‒ you already know it: it's just the expectation value. In our case, it would be the mean temperature. The two-point function tells you something about the correlation length in the distribution.
The relevant quantity we are concerned with here is the three-point function. (Confusingly enough the three-point function is also known as bi-spectrum.) To compute it, you roughly take three different points of your distribution, multiply the value of the function (here the temperature), and integrate over combinations of three points. Even from this rough description you can notice two things. First, it's several ugly integrals that are hard to compute, especially with loads of data. Second, multiplying small numbers makes even smaller numbers, thus the result is in risk of dropping below the uncertainties in your measurements. Therefore it's hard to come by this observable, yet it's what one wants to extract from the data because it contains information beyond the simplest (single-field, slow-roll) inflation scenario. This simplest scenario predicts the temperature fluctuations to be to very good precision a Gaussian distribution. If they were exactly Gaussian, the three-point function would vanish, and all higher-order correlations would follow from the two-point function. A non-vanishing three-point function would thus be, here it comes, an indication for the non-Gaussianity of the temperature fluctuations, and an indication for new physics.
There had indeed previously been rumors that non-Gaussianities had been found in an analysis of the CMB data, see e.g. this post on Non-Gaussian CMB over at Resonaances. At that time I heard like half a dozen talks on the topic, yet was reasonably sure the "signal" would vanish back into noise as indeed it did. To our present knowledge, the data with the uncertainty we have is still compatible with a Gaussian spectrum. (One has to be somewhat careful when one reads about these bounds since there's several different ones. That's because the full three-point function is pretty much impossible to calculate. What people have done instead is to take samples of specific threesomes of points, e.g. those forming equilateral triangles, or obtuse angled ones, thus there's different bounds depending on the triangles chosen.)
However, the important thing to note is that the uncertainty in these observations will go down in the soon future, with the WMAP 8 years mission results one expects a 20% improvement on the bounds, while Planck can yield a factor of 4. Now if there was an indication for non-Gaussianity this would be very exciting. Then the question is of course, what is the physics behind that? What I guess is going to happen is that anybody with their model will predict non-Gaussianities. I wouldn't be surprised if suddenly it will be a signature for cosmic strings, evidence for the multiverse and also a prediction of Loop Quantum Cosmology. It will certainly take some while to sort out these things. In any case however, I am sure it is a topic you will hear more about in the coming years.