It is difficult to explain in a few words the process which creates what we call “classical singing.” We use the word  ‘classic’ to apply to many things, cars, clothing, etc.  But to apply the word classic (or classical) to singing means a complex process that requires quite a lot of detail. I will shorten as much as I can.

Elsewhere in my blogging I have touched on this. To achieve a classically oriented tone quality means use of the vocal mechanism in a particular way.  We singing teachers refer to two types of sound classical singers make, a slightly harsh and strong sound which we make from the middle of the vocal range downward and a soft, “floaty” sound which we make from the middle of the voice upwards. Each of these sounds requires a different muscular usage.  In the operatic/classical song voice, we combine these two unique sounds, using both types of sound together, into a tone which is strong with carrying power (lower voice) and prettiness (upper voice). The power sound most teachers and singers call “chest voice” because it makes us feel vibration in the chest when we use it by itself The sweet upper sound we call “head voice” because the vibration moves higher and the sound of the voice mellows. Power and Prettiness: a good combination to my way of thinking.  When a singer learns to use both of these simultaneously, we create a singing tone which we call classical singing.  To hear this kind of singing, listen to opera singers on YouTube and similar sites.
PS. There are a number of other names singers use for these voice parts but I am avoiding as much verbiage as I can here.