First-order methods use $f(x), \nabla f(x)$ to optimize your function. This contrasts with Second-Order Methods that also use the Hessian $(\nabla ^2 f(x))$ to optimize your function.
While the convergence guarantees of using first-order methods are strong, second-order methods are significantly more powerful. However, the added cost of computing and storing the Hessian matrix typically disqualify these from many use cases — hence why they are not as widespread.