'Red shift' is a key concept for astronomers. The term can be understood literally - the wavelength of the light is stretched, so the light is seen as 'shifted' towards the red part of the spectrum. Something similar happens to sound waves when a source of sound moves relative to an observer.
Redshift and blueshift describe how light shifts toward shorter or longer wavelengths as objects in space (such as stars or galaxies) move closer or farther away from us. The concept is key to charting the universe's expansion. Visible light is a spectrum of colors, which is clear to anyone who has looked at a rainbow.
The cosmological redshift is a redshift caused by the expansion of space. As a result of the Big Bang (the tremendous explosion which marked the beginning of our Universe), the Universe is expanding and most of the galaxies within it are moving away from each other.