Calculus is the mathematical study of change. Central to Calculus is the Limit, which, very loosely speaking, is the value a function gets closer and closer to, as its independent variable gets closer and closer to something else. For example, 1/n gets closer and closer to zero when n gets closer and closer to infinity.

With the concept of the limit in mind, then the Derivative is the instantaneous rate in change of something — that is, the limit of the average rate of change of that thing over h units (think: seconds) as h approaches zero. You can think of the derivative as the instananeous slope (or the slope of the tangent line) of the graph of a function.

Once you understand the derivative, then think about what is in many ways the inverse operation: the Integral is the limit of the sum of a whole bunch of little rates of change in value to get the total value of something. You can think of the integral, or, more properly, the Definite integral as the signed area above the x axis and under the graph of a function between two specific values of x. The Indefinite integral is simply the Anti-derivative. That is, if you find the indefinite integral of a function, and then differentiate it (by the same variable), you get back the original function.

A differential equation is an equation that relates a function to its derivatives. Example: f'(x) = f(x). (This equation is more often denoted $ \frac{\mathrm{d}y}{\mathrm{d}x}=y $, or y' = y, where y is assumed to be a function of some independent variable, typically x.)