Overview
While special relativity constrains objects in the universe from moving faster with respect to one another than the speed of light, there is no such constraint in general relativity. An expanding universe generally has a cosmological horizon which, by analogy with the more familiar horizon caused by the curvature of the Earth's surface, marks the boundary of the part of the universe that an observer can see. Objects beyond the cosmological horizon are moving away so fast that light (or other electromagnetic radiation) is unable to reach the observer.
There are two ways to describe a spacetime with a horizon: locally and globally. The local picture includes only what is (potentially) visible from a given point in spacetime, while the global picture includes unobservable regions beyond the horizon. The two perspectives are related by a process of extension: wherever there is a horizon, a particular solution of General Relativity can be extended beyond it by assuming that nothing special happens there (i.e., that it "looks like" the region within the horizon). The local and global points of view have different notions of time. From the local point of view time stops at the horizon, whereas from the global point of view time extends beyond it, and surfaces of constant time cross the horizon. Ignoring quantum mechanics, the two pictures are equivalent: any statement can be translated freely back and forth.
For cosmology in the global point of view, the observable universe is one causal patch of a much larger unobservable universe; there are parts of the universe which cannot communicate with us yet. These parts of the universe are outside our current cosmological horizon. In the standard hot big bang model, without inflation, the cosmological horizon moves out, bringing new regions into view. As we see these regions for the first time, they look no different from any other region of space we have already seen: they have a background radiation which is at nearly exactly the same temperature as the background radiation of other regions, and their space-time curvature is evolving lock-step with ours. This presents a mystery: how did these new regions know what temperature and curvature they were supposed to have? They couldn't have learned it by getting signals, because they were not in communication with our past light cone before.
Inflation answers this question by postulating that all the regions come from an earlier era with a big vacuum energy, or cosmological constant. A space with a cosmological constant is qualitatively different: instead of moving outward, the cosmological horizon stays put. For any one observer, the distance to the cosmological horizon is constant. With exponentially expanding space, two nearby observers are separated very quickly; so much so, that the distance between them quickly exceeds the limits of communications. In the global point of view, the spatial slices are expanding very fast to cover huge volumes. In the local point of view, things are constantly moving beyond the cosmological horizon, which is a fixed distance away, and everything becomes homogeneous very quickly.
In either view, as the inflationary field slowly relaxes to the vacuum, the cosmological constant goes to zero, and space begins to expand normally. The new regions which come into view during the normal expansion phase, in the global point of view, are exactly the same regions which were pushed out of the horizon during inflation, and so they are necessarily at nearly the same temperature and curvature, because they come from the same little patch of space. In the local point of view, the cosmological horizon still is at the big bang, and inflation is always going on in a thin skin where time is nearly stopped, and the same process produces new regions as it always did, up to small fluctuations.
Inflation from the global point of view is often called eternal inflation. On a global constant-time slice, regions with inflation have an exponentially growing volume, while regions which are not inflating don't. This means that the volume of the inflating part of the universe in the global picture is always unimaginably larger than the part that has stopped inflating. If the probability of different regions is counted by volume, one should expect that inflation will never end, or applying boundary conditions that we exist to observe it, that inflation will end as late as possible. Weighting by volume is unnatural in the local point of view where inflation is not eternal—it eventually ends as seen by any single observer. This picture gives a meaning to the probability distribution on the anthropic landscape.
The theory of inflation in any picture explains why the temperatures and curvatures of different regions are so nearly equal, and it predicts that the total curvature of a space-slice at constant global time is zero. This prediction means that the total ordinary matter, dark matter, and residual vacuum energy in the universe have to add up to the critical density, a prediction which is very accurately confirmed. More strikingly, inflation allows physicists to calculate the minute differences in temperature of different regions from quantum fluctuations during the inflationary era, and these quantitative predictions have also been confirmed.
Read more about this topic: Inflation (cosmology)