Gradient descent uses only local slope. Newton's method for optimization also uses local curvature.

Near the current point $x_k$, approximate the objective by

$$f(x_k+d)\approx f(x_k)+\nabla f(x_k)^Td+\frac12 d^T H_f(x_k)d.$$

Minimizing this quadratic model gives the Newton step $d_k$ from

$$H_f(x_k)d_k=-\nabla f(x_k),$$

followed by

$$x_{k+1}=x_k+d_k$$

or by a shortened step when the full Newton step is not appropriate.

Near a well-behaved strict local minimum, Newton iterations can converge very quickly. Far from a solution, the linear system for the Newton step may fail to determine a useful direction, especially when the Hessian is indefinite.

Second-order methods trade more expensive iterations for richer local information.