If a spring scale measures weight, what does a mass scale use to figure out mass? Are weight and mass measured the same way? 

A spring scale measures weight. It does this by reporting how much upward force it needs to exert on an object to keep that object from accelerating. Since this upward force exactly balances the object's weight (assuming the object isn't accelerating), the upward force reported by the scale is exactly equal to the object's weight. If the scale reports that the object has a certain mass (in kilograms), then it is taking advantage of the fact that, near the earth's surface, each kilogram of mass weighs 9.8 newtons. But it is still measuring weight and using the relationship between mass and weight to determine the object's mass. If you were to move the "mass" scale to a new location, such as the moon's surface, the scale would read incorrectly because the relationship between mass and weight would have changed.