In the field of Cosmology, cosmic inflation is a theory which posits that in the first 10-35 of a second after the Big Bang, the universe underwent a period of rapid, exponential inflation, with it increasing in size by a factor of at least 1026. It was first proposed by Alan Guth of MIT, in 1981. The theory accounts for many objections to the Big Bang model.  
Problems with the Big Bang model
The Flatness problem
The critical density of the universe is the density at which the universe's rate of expansion will slow down, and eventually reach a rate of zero, as the age of the universe becomes very, very large. Densities above or below this would lead to the universe eventually collapsing back on itself during to gravity, or expanding at an every increasing rate, respectively. Today's best estimates of the universe's density place it staggeringly close to the critical density - the universe is described as being almost "flat", since the shape of the universe's geometry is decided by its density - hence the name of the flatness problem. If the universe's density is extrapolated back until just after the Big Bang, it is found to be exceedingly close to 1 - given that it can take any value, this value leading to a perfectly "flat" universe seems unlikely to have been chosen at random. 
The explanation offered by cosmic inflation is that the period of exponential expansion which occurred essentially "ironed out" any small creases in the early universe, leading to the visible universe's shape being flat.
The Horizon problem
In accordance with the Cosmological principle, the observable universe is seen to be both homogeneous and isotropic on large scales (that is, the "same", no matter where you are, or where you look).  Given the vast size of the observable universe, and the finite age of the universe, there is no reason for the universe to be so "smooth" - there has not been enough time for light to travel between two points at opposite ends of the universe, and so they are not causally connected, and there should be no reason for them to have such similar properties.
Cosmic inflation offers an explanation, in that the universe expanded by such a large amount, over such a small time, so the entire visible universe is in fact a very small portion of the early universe (in causal contact with surrounding points at that time) which was expanded into everything seen today.
The Magnetic Monopole problem
Several physical theories, such as General Relativity and some Grand Unified Theories predict the existence of particles called magnetic monopoles, as well as the magnetic dipoles easily observed in nature. However, no monopoles have ever been observed, and given the success of other aspects of the Standard Model, this was unexpected.
However, if the magnetic monopoles were created before the period of expansion posited by cosmic inflation as is expected, they would have been diluted throughout the universe, since the distance between two neighbouring particles would have been increased by such a huge factor, so there may only be one monopole (if that) present in the entire visible universe.