Some of these questions you should be able to find reasonable answers to via wikipedia...
But as to the point of calculus, well there are two at first seemingly different topics in calculus
i) Differentiation
ii) Integration
The first as you say deals with the instantaneous rates of change of functions (and similar objects). For example, a straight line can be described by y(x) = mx +c. Taking the derivative gives dy/dx = m. This is like the `velocity'. Taking the derivative (w.r.t. x) gives zero - the rate of change of the gradient or slope of a straight line is zero, thus it does not change.
There is then an inverse operation of differentiation - given up to an additive constant - this we call the antiderivative. Following the simple example, if I have a constant function m, then the antiderivative is y(x) = mx+c, but the c is arbitrary as there is no way to fix this without some further information.
On to integration, this is usually introduced as a calculational tool for working out the areas under a curve y(x) between x0 and x1 (say). This then can be generalised to find volumes and so on.
The loose idea is to cut up the area under the curve into thin strips and then add up all these strips. For a finite number of strips you get an approximation to this area. If you consider an infinite number of strips then you get the area - but to make sense of this you need limits. This gives us the definite integral as it is between two points.
There is also the indefinite integral, where no bounding points are given.
Now, the amazing thing is the fundamental theorem of calculus tells us that the antiderivative and the indefinite integral are the same thing. Thus differentiation and integration are tightly related.