When you slow down your shutter speed, your sensor records what you aim your camera at for a longer period of time. This means that if something is moving, the movement will be recorded. The effect that we see is called motion blur or implied motion.
In the picture of the waterfall above, my shutter speed was 1/3 of a second. Over the course of that 1/3 second, the water was moving and the rocks were not. Therefore we see the implied motion of the water in the blurry streaks, but the rocks are crisp and clear.
In the picture below, I set my shutter speed to 1/60 of a second to capture the motion and excitement of Belle as she emerged from her hiding place behind the curtain. Notice again, the things which are not moving are sharp, while the things that are moving appear blurry.