Distinguishing between quality and quantity are a convenient way of defining the character of incremental or quantum changes in specific properties of an object. In a very simplistic example usually cited in discussions about dialectics, water being heated undergoes incremental changes in temperature (quantitative change) but its property as a liquid (a quality) does not change. When such quantitative change reaches a CriticalMass, a new kind of change emerges--the water boils and turns to water vapor, which is a qualitative change.
In another way of saying it, quantitative change that proceeds in increments (usually slowly), when it reaches a certain point, gives way to qualitative change (usually abrupt if not explosive). We can also say that the appearance of a certain quality requires a certain quantity. (There's a separate page here, QuantityAffectsQuality, written by another author, that says basically the same thing in a more succinct way.)
The example I gave of water turning to steam is of course a very simplistic example that abstracts from what actually happens from the thermodynamic point of view, as any physics student must be aware of. Concept-words like quality and quantity, when applied to a specific object, must be clearly placed within a well-defined framework of properties.