Observe a motorcycle when it is approaches towards you and when it travels away from you. Notice the sound of its engine. As it comes nearer to you, the buzz gets louder. And as it goes farther, the roar fades. This is basically the idea of the Doppler Effect.
The pitch of the sound of an object gets higher as it approaches an observer and lower as it moves away. In a shorter distance, the sound frequency increases and wavelength decreases. In longer distances, the sound frequency decreases and wavelength increases. Aside from sound waves, this also happens with light and water waves. The Doppler Effect must be first understood to define what a Blue Shift is.
So basically what is a Blue Shift? It is that shortening of the wavelength and increasing in frequency, which means the object is moving toward the observer. It is named blue not because it was the favorite color of the one who discovered it, but because in the optical spectrum (or the rainbow), the blue end has the shorter wavelength. The longer wavelength, the Red Shift then is the increase in wavelength and decrease in frequency, where the object is traveling away from the observer.
It is important in astronomy because celestial things that are moving away, like the Andromeda Galaxy, from the Earth are undergoing a Blue Shift. This way, scientists can know which heavenly bodies are distancing from us.