In the 1940s and 50s, a few scientists (George Gamow, Ralph Alpher and Robert Herman among others) predicted the continued existence of the photons that last scattered in the very early universe. Theoretically, those photons had continued to travel through the universe, cooling as the universe expanded. The early theorists tried to predict what the temperature of these photons would now be (with varying degrees of success). These photons should be all over the place and hence providing a constant "background" to any antenna on earth. In addition, they should have cooled enough that now their wavelength would be in the microwave range. Thus, these photons came to be called the cosmic microwave background.
In the mid 1960s, a group at Princeton led by Robert Dicke began building a radiometer to detect the CMB. At the same time, Arno Penzias and Robert Wilson at Bell Labs observed some noise in a sensitive antenna they were planning to use for radio observation. After careful work, they decided that this noise had to be external and coming from all directions in the sky. Eventually they made contact with the Princeton group, and this background noise was interpreted as being the CMB (after first talking to Penzias and Wilson, Dicke supposedly got off the phone and told his collaborators, "Boys, we've been scooped"). The two groups published companion papers on the observation and the interpretation, and in 1978 Penzias and Wilson received the Nobel Prize.
Although important, that first observation is not on its face all that exciting. The CMB is remarkably smooth or isotropic (meaning it looks the same in all directions). The picture below shows what Penzias and Wilson might have seen if they'd been able to observe the CMB in all directions (courtesy http://map.gsfc.nasa.gov/), and it's hard to see what all the fuss is about. But I'll leave that for the next post.