Short answers, pulled from the story.
Calculus is the mathematical study of continuous change and the principal precursor of modern mathematical analysis. It has two major branches, differential calculus, which studies instantaneous rates of change and slopes of curves, and integral calculus, which studies the accumulation of quantities and areas under or between curves.
Isaac Newton and Gottfried Wilhelm Leibniz each independently formulated infinitesimal calculus in the late 17th century. Newton derived his results first in 1665-1666, but Leibniz published his Nova Methodus pro Maximis et Minimis first in 1684, and Leibniz gave the discipline its name and much of its notation.
The fundamental theorem of calculus states that differentiation and integration are inverse operations, relating the values of antiderivatives to definite integrals. It provides a practical way to compute definite integrals by finding antiderivatives rather than performing limit processes.
In Latin the word calculus means pebble, a meaning that still persists in medicine. Romans used pebbles as counters on their abacuses and called reckoning placing pebbles, and Leibniz is the mathematician who gave the new discipline the name calculus.
Elements of calculus first appeared in ancient Egypt, with volume and area calculations in the Moscow papyrus around 1820 BC, and later in Greece through Eudoxus of Cnidus and Archimedes. The ideas also developed in China with Liu Hui, in the Middle East with Alhazen, and in India with Bhaskara II and the Kerala School.
Calculus is used in physics for all concepts in classical mechanics and electromagnetism, including Newton's second law of motion and the theories of Maxwell and Einstein. It is also applied in chemistry for reaction rates and radioactive decay, in biology for population dynamics, in medicine for drug elimination and tumor growth, and in economics for the Black-Scholes option pricing model.
Infinitesimals were criticized as unrigorous, and Bishop Berkeley famously described them as the ghosts of departed quantities in his 1734 book The Analyst. The work of Cauchy and Weierstrass roughly 150 years later replaced infinitesimals with the epsilon, delta approach to limits, and Abraham Robinson revived them rigorously in the 1960s through non-standard analysis.