Calculus in the modern day has many uses and once a person has learned calculus it is hard to fathom how anything was built successfully without it. Calculus in its simplest terms is the how quickly numbers change. The two mathematicians credited with the creation of Calculus are Isaac Newton and Gottfried Leibniz, though it is argued that if Archimedes had lived a few more years that he would have created calculus a few centuries before either of them were even born.
Archimedes started studying how volume changes and how to calculate volume in a variety of different shapes. Though without the use of calculus his proofs about volume are long and cumbersome. But he did start to discover limits though he called it exhaustion and used it to approximate the area of circles by calculating the area of polygons, with roughly the same diameter, each with more edges than the previous iteration.
In the 17th century the idea of a derivative, the rate of change of a function, came into being. Newton and Leibniz are the main two mathematicians credited with creating calculus. While Newton was the first to come up with calculus, he used it to solve problems in physics, Leibniz was the first to publicize Calculus in its own right. Newton did not publish any papers formalizing calculus instead he just used it in physics problems as can be seen in his correspondence. To figure out the total area he used ratios of changes and showing that each change was infinitesimally small but the number of changes were also infinitesimal.
Leibniz on the other hand started his development of calculus by taking a graph and breaking it up into many small rectangles and then finding the area of these rectangles. Leibniz made many of his discoveries by experimenting with the notation of calculus. He is also who we can thank for the notation behind calculus.
Calculus has been centuries in the making and its development has had contributions from many intelligent mathematicians. In its simplest terms it is the calculations behind the rate of change.