"Red Shift" is a measure of the Doppler Effect. Light waves have are more red when they have a lower frequency and more blue when they have a higher frequency. If the source of a wave is moving away from us (e.g., a distant star or galaxy), the wave frequency of its light appears lower. Although sound waves are very different from light waves in space, it is a good analogy to think of how a siren sounds when it is approaching: The pitch is higher (higher frequency sound waves) as it approaches us, but after it passes, the pitch decreases. This is the Doppler Effect for sound waves.
When we talk about a celestial object being "red shifted" we mean it is travelling away from us. Because we observe a red shift in the distant galaxies, it is evidence that the universe is expanding.