The horizon problem is a problem with the standard cosmological model of the Big Bang which was identified in the 1970s. It points out that different regions of the universe have not "contacted" each other due to the great distances between them, but nevertheless they have the same temperature and other physical properties. This should not be possible, given that the exchange of information (or energy, heat, etc.) can only take place at the speed of light. The horizon problem may have been answered by inflationary theory, and is one of the reasons for that theory's formation.
Here is their propossed solution.
Inflationary theory allows for a solution to the problem (along with several others such as the flatness problem) by positing a short 10 − 32 second period of exponential expansion (dubbed "inflation") within the first minute or so of the history of the universe. During inflation, the universe would have increased in size by an enormous factor.
If correct, inflation solves the horizon problem by suggesting that prior to the inflationary period the entire universe was causally connected, and it was during this period that the physical properties evened out. Inflation then expanded it rapidly, freezing in these properties all over the sky; at this point the universe would be forced to be almost perfectly homogeneous, as the information needed to change it from that state was no longer causally connected. In the modern era distant areas in the sky appear to be unconnected causally, but in fact were much closer together in the past.
One consequence of cosmic inflation is that the anisotropies in the Big Bang are reduced but not entirely eliminated. Differences in the temperature of the cosmic background are smoothed by cosmic inflation, but they still exist. The theory predicts a spectrum for the anisotropies in the microwave background which is mostly consistent with observations from WMAP and COBE.
I don't know very much about these issues and I would like to hear everyones input.