If you've ever taken your point and shoot to a dark room and ended up with a grainy picture, you've experience with image noise.
Image noise is, to be short, the grainy look that images get. Noise is affected by both ISO
and the size of the image sensor, but since the sizes of most DSLRs are the same, the biggest factor for image noise is ISO.
ISO is a measurement of how sensitive the image sensor is. The higher the ISO, the more sensitive the image sensor and the less light (smaller aperture or faster shutter speed) necessary for a proper exposure. Unfortunately, cameras make the sensor more sensitive by running more electricity through it, which can lead to the pixels becoming hot, resulting in that grainy look.
To show the difference, we now turn back to the photomajig, now with the image cropped to highlight the image noise:
As you can see, while you can start to make out the noise at ISO400, it's extremely noticeable at ISO3200. Keep in mind that newer DSLRs can handle noise better @ ISO3200, and full frame cameras can chew up that ISO and will pop out a clean image, but the principle is the same: high ISO = more noise.