The word "calculus" literally means "pebble" or "stone" (from the same root as calcium, and ultimately, chalk), it is still used with this meaning in medicine. Originally, these calculi were used for counting (or in more sophisticated form, in an abacus). Thus calculus came to take on the meaning of the counting operations one performs in arithmetic.
Thus "a calculus" was a systemic way of performing calculations. The official name of what is now (as a math subject) called calculus, was: The Differential and Integral Calculus. This was shortened to "the calculus" as a way of distinguishing it from other calculi, for example synthetic division of polynomials. Although initial resistance to *this* calculus was high, it soon gained central prominence because of its wide application to problems involving force and motion, basically laying the technical groundwork for the machinery that fueled the industrial revolution. Physics, in particular, grew by leaps and bounds as this new tool was aggressively applied to many long-standing problems in mechanics, astronomy, optics, thermodynamics, and electricity and magnetism.
As is human tendency, the shortened form stuck, with the longer formal name implied thereby. For quite a long time, it stood as what was considered the pinnacle of human mathematical achievement, a sign that perhaps we really would one day unlock all the secrets of the universe (humans tend to do this, too- we really believe "we're all that"). As other calculi faded into obscurity (algebra was regarded for a long time as just a pedestrian calculation tool, something to master only so one could use it in service to "the calculus"), the "the" was also dropped, and "the calculus" simply became "calculus", or even just "calc" (such as in: "I failed calc this semester, the prof was a toad.").
By contrast, other branches of mathematics are often regarded as "methods" (the word "algebra" originally *meant* method, or reckoning, a cognate of the word "algorithm" used in computer science). So we have several "flavors" of algebra, or topology, or even geometry, none of which is distinguished enough from the rest to be called "the" (except, sometimes, in the sense of "the usual").
So "the calculus" becomes a somewhat archaic usage in today's parlance, although still used, because it really did usher in a revolutionary role for mathematics: not as something used to investigate knowledge, but as something which might serve as the basis for knowledge itself: mathematical proofs are regarded (even by laymen) as some of the more indisputable methods of demonstration available to human reasoning. There are those who believe, rather ardently, that the structure of everything we see has a mathematical basis (which we may, or may not, be able to decipher).
