The CBR and Inflation:
By accident, in 1965, two Bell Labs technicians Penzias and Wilson discovered the Cosmic Background Radiation while testing a microwave reciever for a satellite communications experiment.  No matter where they pointed their antenna, which was shaped like the section of a sphere (a 'horn'), they recieved a faint signal at 7.35 cm wavelength.  The fact that the direction of the antenna did not matter indicated that the source of the radiation was ubiquitous.  Meanwhile, as the technicians struggled to find the source of the radiation, a group at Princeton university had predicted a cosmic background radiation but had not yet physically searched for it.  The signature temperature of the fluke microwave signal matched the temperature predicted by the Princeton group, about 3 degrees Kelvin.
Through the 1960's and 1970's more accurate studies were done, at varying wavelengths (the Penzias Wilson measurement was of only one wavelength) to confirm that the 3K source was indeed a 'black body' emmitter whose source was the Big Bang.  The Cosmic Background Explorer satellite confirms a black-body source of 2.7 K.
The radiation in the cosmic background is from a decoupling event, some 300,000 years after the Big Bang, when space became transparent to matter.  The photons produced by the previous set of nuclear interactions are in surplus when space becomes cool enough for protons and other sub-atomic particles to exist.  These photons form the background radiation, distorted by our frame of refrence here in the present, so that what once was a 3000 K characteristic temerature is measured by us to be 3 K.

COBE map from 1990

The Inflationary epoch, proposed by Allan Guth in the 1980's, holds that during the first second of the universe's existance it underwent an exponential expansion, much faster than the current rate, to drive apart the matter that would have else collapsed under its own gravitational field.  Still controversial but probably necessary for a complete cosmological theory, inflation holds that during the time between 10E-35 and 10E-30 seconds after the big bang, the universe expanded exponentially by a factor of 10E50!