"What do we mean by the Big Bang theory?"
The Big Bang theory is the theory that all we can see began as an "incredibly tiny, hot, and dense collection of matter and radiation" (p634). This collection began cooling and expanding when the universe was 10^-43 seconds old ("Planck time"). Current theories can't describe what took place in the Planck era, the time between the Big Bang and Planck time.
"Briefly describe the two key pieces of evidence that support the Big Bang theory."
The first of the two key pieces of evidence for the Big Bang theory is cosmic microwave background that fills the universe. It matches the Big Bang's prediction that radiation began to stream across the universe after the end of the age of nuclei.
The second key piece is the observation that the universe's helium content matches that predicted by the Big Bang. Stellar fusion would only produce 10% of what is observed, but the Big Bang theory accounts for the rest by saying some of the universe's original hydrogen would have fused into helium in the age of nucleosynthesis.
"Briefly describe how the cosmic microwave background was discovered."
The cosmic microwave background was discovered in 1965 by Arno Penzias and Robert Wilson of Bell Laboratories in New Jersey. Each measurement made with a sensitive microwave antenna they were calibrating for satellite communication was showing unexpected noise. The noise came from all directions in the sky and proved impossible to eliminate. Around the same time, a group at Princeton decided that radiation from the Big Bang should be permeating the universe and therefore be detectable with a microwave antenna. A chance meeting between Penzias and an astronomer from Princeton led to the identification of Penzias and Wilson's noise as the cosmic microwave background and a Nobel Prize in 1978 for these two physicists.
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