Unfortunately, the first question most beginners ask is not "What is this telescope’s aperture?" but "What is its magnifying power?" The truth is, any telescope can be made to provide almost any magnification, depending on what eyepiece is used. The factor that limits the highest power that can be used effectively on a given scope is, you may have guessed, its aperture. As magnification is increased, and the image in the scope grows larger, the light gathered by the telescope is spread over a larger area, so the image is dimmed. There is also an absolute limit, determined by the physical properties of light, to the resolution that is possible with any given aperture. As the magnification is pushed beyond that limit the image fails to reveal any additional detail and gradually breaks down into a dim, fuzzy blob.
The maximum useful magnification for any telescope is about 50 times the aperture in inches, or two times the aperture in millimeters. This equates to about 100x to 120x with the smallest telescopes, which is enough to see such wonders as the rings of Saturn and cloud bands on Jupiter. The 2x per millimeter figure is a rule of thumb, and can vary up or down somewhat depending on the optical quality of the scope in question and the vision of the individual observer. Experienced observers usually use much less power; 0.5x to 1x per millimeter is more appropriate for most objects. Any manufacturer claiming that their 60mm scope can provide good views at 450x (7.5 times the aperture in millimeters) is trying either to pull your leg or pick your pocket!