Its more about a period in time and the style of music relating to that period than about 'class'.
http://en.wikipedia.org/wiki/Classical_period_(music)
In the middle of the 18th century , Europe began to move towards a new style in architecture, literature, and the arts, generally known as Classicism, which sought to emulate the ideals of Classical antiquity and especially those of Classical Greece.[1] While still tightly linked to the court culture and absolutism, with its formality and emphasis on order and hierarchy, the new style was also a cleaner style —one that favored clearer divisions between parts, brighter contrasts and colors, and simplicity rather than complexity. The remarkable development of ideas in "natural philosophy" had established itself in the public consciousness with Newton's physics taken as a paradigm: structures should be well-founded in axioms and be both well-articulated and orderly. This taste for structural clarity worked its way into the world of music, moving away from the layered polyphony of the Baroque period, towards a style where a melody over a subordinate harmony —a combination called homophony— was preferred. [2] This meant that the playing of chords, even if they interrupted the melodic smoothness of a single part, became a much more prevalent feature of music. This, in turn, made the tonal structure of works more audible.
As for pop
The new style was also pushed forward by changes in the economic order and in social structure. As the 18th century progressed, the nobility became the primary patrons of instrumental music, and there was a rise in the public taste for comic opera. This led to changes in the way music was performed, the most crucial of which was the move to standard instrumental groups and the reduction in the importance of the continuo — the harmonic fill beneath the music, often played by several instruments. One way to trace this decline of the continuo and its figured chords is to examine the decline of the term obbligato, meaning a mandatory instrumental part in a work of chamber music.
The nobility paid for classical music, and they paid for what they liked. It was pop as far as they were concerned.
In terms of style of music I've always preferred 'Baroque' over 'Classical', not that I claim any detailed knowledge of either.
