"Newton's method" is a numerical method for solving an equation that basically replaces the function at a given value of x by the tangent function at that point. Here the function is f(x)= x^3- 8 which has derivative f'(x)= 3x^2, the derivative at x_0 f(x_0)= 3x_0^2 while the value of the function is x_0^3- 8. So the tangent function at x= x_0 is y= 3x_0^2(x- x_0)+ x_0^3- 8. Setting that equal to 0, 3x_0^2(x- x_0)+ x_0^3- 8= 0, 3x_0^2(x- x_0)= 8- x_0^3, x- x_0= \frac{8- x_0^3}{3x_0^2}, and x= x_0+ \frac{8- x_0^3}{3x_0^2}.
Start with some reasonable value for x_0 and calculate the next value for x: with, say, x_0= 1, x= 1+ \frac{8- 1}{3}= 1+ \frac{7}{3}= \frac{10}{3}. Now take x_0= \frac{10}{3} and calculate the next value for x. Repeat until you get two consecutive values for x that are closer together than your allowable error.
x_0= 0 is not a "good" starting value (in fact, it is impossible) because the denominator, 3x_0^2, would be 0.
