Here is a picture of the electromagnetic spectrum.
What it basically shows, is that the length of a light wave determines what kind of light it is.
Waves have this one characteristic, they stretch or compress when they follow an object that moves at a speed comparable to the speed of the wave.
When sound waves stretch, we can hear this stretching. When a police car with a siren approaches us, passes us, and moves away from us, we can hear the pitch of the sound changing. That is due to sound waves contracting and than stretching as the distance between us and the car decreases, than increases. As the car moves away from us, the increase in distance from us to the car causes the sound wave to stretch.
To picture it better, imagine a wave between two points. If one point remains in the same place, and another point to which the wave is attached, moves away, the wave is stretched.
Same thing happens to light waves. Except with light, different ranges of light waves produce different types of light. If you look at the diagram above, you will see that each color and type of light has a specific lightwave range. A lightwave is the distance between two consecutive peaks of a wave.
In the visible light range, blue light has the shortest wavelengths, and red light has the longest wavelengths. Stretch a blue wave of light far enough, and it will become red light. So, the more you stretch it, the closer to red it gets. That is called a red-shift.
So, red-shift simply refers to the phenomenon when a wave of light gets longer, and hence closer to been red, as an object producing it, moves away.