Generally they're denoted "A-BxC" for variable power scopes. "A" is the lowest magnification setting, "B" is the highest, and "C" is the diameter of the objective lens (front lens) in millimeters. Fixed power scopes don't have the second magnification number, meaning they look like this: "AxC," where "A" is the magnification and "C" is, once again the objective lens diameter. Thus, a 3-9x40 will have an objective lens 40mm across and will make a target that subtends 1 minute of angle appear to subtend 3-9 minutes of angle, depending on what setting the shooter has chosen.
The most common tube diameters (the width of the narrowest portion of the scope) are 1 inch and 30mm. While 1 inch will handle the needs of most shooters, 30mm is purported to allow more light through to the eye, allowing better vision in low light.
Another measurement is the parallax setting. I'd wager the bulk of centerfire scopes on the market are set to be parallax free at 100 yards, though shotgun scopes may be set for 50 or 75. Adjustable objectives are handy for shooting at a variety of ranges.
Depending on what type of shooting you're looking to do, 3-9x40 (or thereabout) is a pretty standard jack-of-all-trades. Higher powers may work better for target shooting while lower powers are more suited on woods guns.
