What is the Doppler shift, and why is it important to astronomers?

The Doppler Shift is the same thing as the Doppler effect.

The doppler effect just states that, to an observer, a sound appears to sound higher in pitch (frequency) as it gets closer to them and lower in pitch (frequency) as it moves farther away.

Think of a fire truck or police vehicle siren getting very high pitched right as it passes by you, and then getting lower in pitch as it drives away from you.

Astronomers can use this because you can use emitted sounds to calculate the distance or velocity of an object.