The area of math that I have most enjoyed so far is Differential Calculus. When taking an early algebra class we would discover the slope of a line. I always thought that was cool at the time. However, being able to view a polynomial function on a graph and finding the slope of that line is much more gratifying.
Wikipedia of course has a very detailed explanation to Differential Calculus at http://en.wikipedia.org/wiki/Differential_calculus.
Derivatives are used to calculate the slope of a function at a given point. They can be very simple at times like when finding the derivative of $x^2+2$ the derivative is $2x$. For example, the slope of this parabola at $x=3$ is $6$. All we had to do was plug in the point at which we are wanting to calculate the slope of our function.
The most basic way to calculate a derivative is to use what is called the Power Rule.
Derivatives can be used for many different things. As I said before they can tell you the slope, but also they are used to find the maximum or minimum of a function, and can also let you know the curvature.