How are the expanding universe and the Doppler effect related to one another?
The Doppler Effect is evidence that the universe is expanding. When stars or galaxies are moving away from us, we see their color as red shifted. Because the distant galaxies are red shifted, this means that they are all receding with respect to Earth. In the theory of Special Relativity, Albert Einstein predicted that spacetime itself is expanding.
The observation of the Doppler Effect on light was a problem for physics because the frequency of light does not change. Einstein's famous conclusion was that time was changing with respect to gravity, while the velocity of light remained constant.