x_new = x - f(x)/f'(x). Quadratic convergence roughly doubles the number of correct digits each step.
The standard finance application is backing out implied volatility, where the derivative is vega and is available analytically.
The weakness: it needs a good starting point and can diverge, oscillate, or run away if the derivative is near zero. Production implementations pair it with bisection - slow but guaranteed to converge if the root is bracketed - and fall back when Newton misbehaves. Being able to say why you would combine them is the complete answer.