Inflation

Inflation is one of the most popular methods known for generating an isotropic and homogeneous universe. The basic idea is that the universe, which is expanding now with some time dependent scale factor a(t), starts expanding faster and faster. Expansion means that the scale factor changes with time, d a(t)/dt is nonzero. For inflation, this change in the scale factor changes with time as well, that is, the scale factor accelerates:

d² a(t)/ d t² > 0 .

The effect of this is to dilute away all the inhomogeneities in space, such as monopoles predicted by many high energy theories (one of the early motivations), or any other sort of fluctuations. The number density of particles goes to zero, since space is expanding so fast.

Horizons
A special feature of inflation is its effect on horizons. The horizon demarcates the boundary of causally connected regions, regions that light rays (which travel at the fastest speed that any signal can travel) can reach since the time of the big bang. These regions grow over time, as light has more time to travel, but the expansion of the universe means that over time there is more space to cross as well. When the universe isn't inflating, such as now, regions which are larger and larger come inside the horizon and become causally connected. During inflation, the expansion of the universe wins out. Regions which were causally connected are separated so fast by the expansion of space that a region once in causal contact can have parts of it "pushed out" of the horizon. Thus we expect scales larger than our horizon would not be correlated (that is, were once able to affect each other causally) unless inflation occurred. For example, the Cosmic Microwave background is homogeneous on scales which were not in causal contact when the signal was created, and thus suggests we need something like inflation to explain its homogeneity.

What's Left?
During inflation, the only reason that space doesn't become completely boring is that "empty space" can have energy (and does during inflation, as will be seen below) and that "empty space" also always has quantum fluctuations. These quantum fluctuations can provide the seeds for the formation of structure after inflation ends.

Inflation can end when the energy in "empty space" goes back into (kinetic) energy for the fields in space, turning into a bath of particles (`reheating'). This change from being the energy of space to kinetic energy of particles arises very naturally when particles are treated as the quantum fields that they are. In some scenarios inflation only ends in some regions of the universe and parts of the universe continue to inflate forever.

For pictures and another discussion of inflation, see this page by Ned Wright.

In more (mathematical) detail
Consider a spacetime with some fluctuations superposed on a homogeneous space background-- the background metric is
ds² = dt² - a²(t) d x²
where

t is time
a(t) is the scale factor for the expanding universe
x refers to the spatial coordinates
"dx²" represents the homogeneous space of positive, negative or zero (flat) curvature k, i.e. with k = +1, -1 or 0.

So this space is changing in time with scale factor a(t).

The matter in spacetime affects the space and space affects the matter via Einstein's equations. In the presence of matter with density rho and pressure p (fluid form), and vacuum energy lambda (cosmological constant), Einstein's equations for the scale factor become

(da(t)/dt)² /a(t)² = H² = 8 Pi G rho/3 - k/a(t)² + lambda/3

The properties of the matter enter here through the density rho.

One way to get inflation, an accelerating scale factor a(t), can be seen with a toy model. Consider a very simple world where there is only the metric (with scale factor a(t)) above) and some matter represented by a scalar field phi. Normally the field can have kinetic and potential energy. Take the potential energy of phi, V(phi), approximately constant. Since V(phi) is a function of phi, this doesn't have to be true for all values of phi, say just for some values of phi, and take the kinetic energy a lot smaller than the potential energy. Then the total energy rho will be

rho = kinetic energy + V(phi): ~ V(phi) ~constant

by assumption.
Using this expression for the energy density in the equation for the metric, from above,

(da(t)/dt)² /a(t)² = 8 Pi G constant/3 - k/a(t)² + lambda/3

and as a(t) increases, we eventually get

(da(t)/dt)² /a(t)² ~ C

where C = 8 Pi G constant/3 + lambda/3, a time independent constant.

Thus this equation is easy to solve, to get

a(t) = e^{C^1/2 t}

That is, exponential expansion.

In field theory, you specify the equations of motions for the field, and can get the example of
potential energy V >> kinetic energy in a few ways. The kinetic energy is related to the change in the potential with the field, so in the simple toy models often used for examples, you can be at an extremum of the potential or just have the potential change very slowly with phi ("slow roll").

So if there is just one field phi dominating, plus gravity, inflation can occur if
dV(phi)/d phi is "small"
for whatever value(s) of phi describes the universe at that time. There is a quantitative description of how small "small" must be for a given theory.

Directions
Many different areas are under study in the inflationary scenario. Some are, for example,

Reconstruction. Characterizing different inflationary potentials is of interest because upcoming experiments will be able to measure some properties of the inflaton potential. If many regions are still inflating now, one can try to calculate the probability for being at a region of the universe with our history.
Motivation from Particle Theory. High energy particle theory should in principle describe physics at the scales where inflation might occur. Thus the 'toy model' potentials should be derivable from these high energy theories, which in turn may perhaps give information on which potentials are possible.
Open Inflation. Inflation in its simplest form makes the universe flat as well as homogeneous, but there are some recent suggestions for how an open universe may come about as well. Currently this is strongly disfavored by the data. The front running theory is a flat universe with a cosmological constant about twice the size of the matter density today.
Reheating/ back reaction. There has recently been increased understanding of the behavior of the universe as the energy in the vacuum goes back into particles. The newer approaches study not only the background energy going into particle production, but also the back reaction of this production on the background. Back reaction effects during the inflationary period have been studied recently as well.
Quantum cosmology. One might ask what the initial conditions were for inflation, or how the universe behaved before inflation. More generally quantum cosmology is concerned with the initial conditions of the universe. Some related web pages are at Cambridge University and UCSB, and there are some introductory lectures at a technical level at this site.