Calculus is the incremental study of the continuous. Most everything prior to calculus studies the material in a "discrete" way, i.e. by analyzing individual key points of data. The fundamental difference between calculus and pre-calculus is the concept of the limit. Here is a link to a written description of a limit:

http://www.themathpage.com/aCalc/limits.htm

It can get bogged down in technical concepts, if you have the time, here is a link to a series of cartoons that will help clarify the concept of a limit:

http://www.calculus-help.com/funstuff/tutorials/limits/limit01.html

Now, calculus itself comes in two flavors, differential and integral. Differential calculus studies a notion called the derivative. A derivative is actually very simple. It is defined as the slope of the tangent line to a curve. It turns out the derivative has a lot of practical interpretations. Here's a list of a few:

http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeInterp.aspx

Arguably, the most important interpretation is called a rate of change. For example, the derivative of a position function (where something is as a function of time) can be interpreted as the velocity of that something. (Velocity is a specific example of a rate of change.)

Most large corporate businesses base their decisions on "margins". Which is why you might see a company layoff a division of employees even though the division is making money. The corporation may decide the margins are not large enough. A margin is the economists word for derivative. They are the same thing.

Integral calculus studies the area below a curve. More specifically, the area trapped between a function and the x-axis. Once again, depending on what type of curve we are looking at, this area has a physical interpretation. Suppose we consider a curve that represents a city's population growth over time. The area under the curve would represent the total number of people in the city (or the total increase in population over a time interval).

As another example, If we graph a businesses total revenue on the same plot as its total expenses versus time, the area between the two curves then represents the total profit.

Newton and Liebnitz are both given credit as "discovering" calculus, but many of the notions were known prior to their work. The reason these two are given credit for discovering the field is that they both proved the fundamental theorem of calculus which, in a nutshell, states that these two branches are two halves of the same coin. The theorem can get technical if you don't know what you're doing, but here's a couple of links to the theorem:

http://www.math.hmc.edu/calculus/tutorials/fundamental_thm/

http://www.sosmath.com/calculus/integ/integ03/integ03.html

If you want a few more practical applications here are a few:

http://www.intmath.com/Applications-integration/Applications-integrals-intro.php

Calculus is a branch of mathematics involving derviatives, integrals, limits, series and functions. The two branches of caculus are differential calculus and integral caclulus. In differential calculus the properties and applications of the derivative are used. A derivative can be thought of as a rate of change or the slope of a curve at a given point. The process of obtaining the derivative is called differentiation. For example, if you have a parabola in the form y= x-sqaured and you want to find the slope of the parabola when x = 3, take the derivative of x-squared. That is found by multiplying the exponent of 2 by the coefficient (the number in front of the x-squared).. That product is 2 which now becomes the coefficient. Subtract 1 from the exponent to obtain the new exponent of the derivative. So the derivative of x-square is 2x. Now to get the slope at x=3, put 3 into the equation of the derivative to get 2(3)= 6. So the slope of the parabola (rate of change of y divided by the rate of change of x) at x=3 is 6. Finding this slope enables one to obtain the equation of the tangent line to the curve at any given point. Derivatives are also used to find the normal to a curve. There are many other ways to get derivatives, (product rule, quotient rule, u-substitution method, etc), but giving a basic technique with the parabola problem.

In integral calculus the properties and applications of integration are studied. In geometry one can find the area of triangles, rectangles, pentagons, etc. One use of integral calculus is to find the area under a curve between two points. Take the parabola y=x-squared, draw the curve and say you want the area under that curve between x=1 and x=3. When integrating you take 1/(exponent +1), which now becomes the coefficient and you add 1 to the exponent.. So integration gives (1/3)x^3. Substitue the value of x=3 to get (1/3)(27)= 9 and subtract off what you get when substituting x=1, which is (1/3)(1)= 1/3.. So the area is 9-1/3 = 8 2/3. Integration is also used to find the volume of solids and there are also many other methods used to find integrals, the basic technique is shown with the parabola example.

Calculus is used in various fields of science, econonimcs, medicine and engineering. In economics, it is used to find marginal cost and revenue, which is used to calculate maximum profit. In physics it's used to find, mass, moment of inertia and energy of objects. It's also used to calculate heat transfer with use of Fourier Series. Various methods of calculus are used to appoximate solutions of equations. Another application is finding the velocity, acceleration of a car, bus, train and average altitude of an airplane.

Calculus is a fundamental branch of mathematics.

Calculus was really useful when Newton applied it to figure out the orbit of planets as well as their volume. This is the use in applied physics and, without it, we would not be able to travel into space.

Newtonian physics, Economics, Engineering, any science that deals with probability.