Calculus is the study of change and accumulation. It forms the foundation and context for most forms of modern scientific thought. Differential Calculus introduces a general tool--the derivative--for understanding and quantifying change as a measurable thing. For position, there is velocity, a quantifiable (i.e., with units), measure of how position changes. For anything else (volume?), there is a derivative, which is to it as velocity is to position. Differential Calculus allows you to calculate these quantities and relate them to one another. Typically, if high-school math can solve a problem in which things are constant or change in simple proportions, then calculus can solve the same problem when things begin to change in complicated, curvy, nonlinear ways. A typical problem of differential calculus might look like this:

A hemispherical bowl with radius 3 inches is half full, with water constantly entering at 3 cubic inches per second. How fast is the water level rising?

Integral calculus is the study of how things accumulate over time or distance. A typical example of integral calculus might be this question:

If sunlight falls on my solar collector with an intensity of f(t) = 1/(1+t^2) [or whatever function], how much solar power will I collect in one day?

What is the volume of a bowl manufactured by rotating the parabola y=x^2 around the y-axis?

What is the circumference of an ellipse with major radius 6 and minor radius 5?

Differential and Integral Calculus study rates of change and accumulations, respectively. They are taught in the same course because of one (two?) simple theorem(s), namely the Fundamental Theorem of Calculus. I guess you'll have to take the course to learn about that.