B. The basic idea
Inflation is based on the idea that the Universe could have been dominated at very early times by an unusual type of matter with an equation of state p = w , with w < - 1/3. In most models, these conditions are achieved by a scalar field (the "inflaton") which enters into a state where the potential energy density (V()) dominates over other terms in the stress-energy tensor. Under these conditions, the inflaton has an equation of state with w close to -1, and the Universe enters a period of quasi-exponential expansion called inflation. In most models, the inflaton evolves classically down the potential V; these are called "slow roll" models.
During inflation, the spatial curvature becomes negligible, leading to a "flat" universe (with tot = 1). Also, the quasi-exponential expansion pushes field modes from infinitesimal scales all the way to the size of the observed Universe and even well beyond that. Specific calculations allow us to follow the "zero point" quantum fluctuations in these modes out to cosmic scales and lead to concrete predictions for the primordial perturbations produced by a given inflationary scenario.
Cosmic Inflation has predictive power because details of the state of the Universe before inflation are hidden beyond the domain of realistic observations. The observable features of the Universe after inflation are specified by the dynamics of inflation and are insensitive to the initial conditions. To realize this picture, a minimum number (Ne) of e-foldings of the scale factor during inflation must be achieved (for example, Ne 60 for inflation at the Grand Unification scale). After a sufficient period of inflation, energy must be transferred from the (dominant) inflaton field into ordinary matter via inflaton decay, causing the Universe to "reheat".
A crucial aspect of the inflationary scenario is that it radically changes the causal structure of the Universe as compared with the Standard Big Bang. It is only thanks to these changes that one can hope to explain the state of the Universe using causal processes. Thus, inflation is noted for "solving the horizon problem" (in the sense that it makes a Universe that appears homogeneous over the present Hubble scale much more probable, given a variety of initial conditions), in addition to explaining specific features of the observed Universe.