A photograph, whether it is recorded digitally or on traditional film, is usually regarded as a frozen instant in time, but this is not strictly true. It is a record of a very short interval of time, but it is not instantaneous. The time interval recorded is the shutter speed used to take the photo. In most cases this is in the order of less than 1/100th of a second, which is quick enough to freeze most movement, but in some circumstances, such as in low light conditions or when using narrow apertures, it is necessary to use slower shutter speeds, and this is when movement blur becomes a problem. When using a slower shutter speed, not only will any movement by the subject be blurred, but also any movement of the camera while the shutter is open will cause the entire picture to be blurred. This is even more of a problem when using longer focal lengths, such as with a zoom lens at its longest setting. The lens not only magnifies the subject, it also magnifies any movement, exacerbating the effects of camera shake.
No matter how rock-steady you think your hands are, it is impossible to reliably hold a camera steady enough to take shake-free photographs at slow shutter speeds. There is a rule-of-thumb to approximate the minimum shutter speed for shake-free hand-held shots. It is approximately the reciprocal of the focal length of the lens in millimetres, so if you're using a lens with a focal length of 40mm (the wide-angle setting on many 3x zoom digital compacts is equivalent to about 38mm) you shouldn't use a shutter speed slower than 1/40th of a second for hand-held shots. Likewise if you're using a 120mm lens (the telephoto end of most 3x zooms is about 115mm) then a shutter speed of 1/120th of a second is required for shake-free shots. When you realise that many modern super-zoom cameras have telephoto settings longer than 400mm you can see that this can be a real problem. Not surprisingly, image stabilisation was first introduced on super-zoom cameras and long telephoto SLR lenses.