What exactly is calculus? I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:


  What exactly is calculus? 


I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I heard it was used to calculate the area under a curve on a graph. If anyone can help me with this, I'd much appreciate it.
In a nutshell, Calculus (as seen in most basic undergraduate courses) is the study of change and behaviour of functions and sequences. The three main points are:


Limits: How sequences and functions behave when getting closer and closer to a desired point (geometrically, what happens when you "zoom in" near a point)
Derivatives: How functions change over a parameter (geometrically, the "slope of a graph at a given point")
Integrals: What's the cumulative effect of a function (geometrically, the "area under a graph")


And obviously (and maybe especially), how these relate to one another; the crowning jewel of Calculus is probably the Fundamental Theorem of Calculus, which truly lives up to its name and was developed by none other than Leibniz and Newton.
It's too simple. The derivative of a function gives its rate of change at a given point. The integral of a function gives the area under the curve at a given point. The fundamental theorem merely says that the rate of change of the area under the curve is the value of the function at a given point.

In other words, the derivative of the integral of a function is the function itself. Voila!