Newton's method is a technique for solving equations of the form
$f(x)=0$ by successive approximation. The idea is to pick an initial
guess $x_0$ such that $f(x_0)$ is reasonably close to 0. We then
find the equation of the line tangent to $y=f(x)$ at $x=x_0$ and follow it
back to the $x$ axis at a new (and improved!) guess $x_1$.
The formula for this is
$$ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}.$$
We then
find the equation of the line tangent to $y=f(x)$ at $x=x_1$ and follow
it back to the $x$ axis to get a new (and improved!) guess $x_2$ from
the formula
$$ x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}.$$

We keep on refining our guesses until we are close enough for whatever
application we have in mind. In general, we have the recursive formula

$$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

In typical situations, Newton's method homes in on the answer
extremely quickly, roughly doubling the number of decimal points
in each round. So if your original guess is good to one decimal place,
5 rounds later you will have an answer good to 30+ digits.