In practical usage, the larger the focal length, the more magnification the lens produces. (Gallery of example images at different focal lengths)
For point and shoot cameras, the notations such as "10x" or "15x," relate to focal length. Even a camera that says "15x" has a lens with an actual zoom range just like the larger SLRs. If you look in the technical specifications of the camera, there will be a listing for the lens strength equivalent to an SLR lens. For example, a camera with "15x" marked on the box might actually have a 21mm-315mm lens. The 15x refers to the multiplication of the smallest number of the lens (widest field of view) to get the strongest magnification the lens is capable of recording. Another camera with 15x on the box might have a 35-525mm lens. Unfortunately, there is no way to tell what the 15x actually translate to until you read the technical specifications on the lens.