Calculus is the mathematics of things that change. Algebra handles quantities that sit still; calculus handles quantities in motion: a falling stone, a growing population, the area swept out by a curve. It grew up in the seventeenth century, in the hands of Newton and Leibniz, to answer two questions that ordinary algebra could not.
The two big ideas
The first question is about rates. If you know where a car is at every instant, how fast is it going right now? Average speed over an hour is easy (distance over time) but "right now" is an instant, and dividing by zero time is nonsense. The derivative is the tool that makes sense of an instantaneous rate of change.
The second question is about totals. If you know how fast the car is going at every instant, how far does it travel over the whole trip? You are adding up infinitely many tiny contributions, one for each instant. The integral is the tool that adds them up.
Two sides of one coin
These look like opposite problems (one takes things apart, the other puts them together) and that is exactly what they are. Speeding up from a position, then adding those speeds back up, returns you to the position. This near-magical fact, that differentiation and integration undo each other, is the Fundamental Theorem of Calculus, and it is where this course is headed.
What you already need
Everything here rests on one idea we will make precise next: the limit, the value a quantity approaches as you slide toward some point. With limits in hand, the derivative is a limit of slopes and the integral is a limit of sums. Keep a picture in your head throughout: a curve on a graph. The derivative is its steepness at a point; the integral is the area beneath it. Almost everything else is detail.