According to wikipedia:

In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a

method for finding successively better approximations to the zeroes (or roots) of a real-valued function.

It means, it is beeing used to solve solution of this equation: f(x) = 0.

We have to know derivative f '(x)!

How it works?

1. the algorithm starts with a guess  X0.

2. we calculate X1 = X0 - ( f(X0) / f '(X0) ) - Geometrically, x1 is the intersection point of the tangent line to the graph of f, with the x-axis. The process is repeated until a sufficiently accurate value is reached.

3. in loop calculate Xn+1 = Xn - ( f(Xn) / f '(Xn) ). Repeat until accurate value is reached.

You can download an example of the calculation in XLS document here.