Newton's Method is a process by which we estimate the roots of a real-valued function. You may remember the bisection method, where we find a root by creating smaller and smaller intervals. Newton's Method uses the derivative in order to account for both the speed at which the function changes and its actual position. This creates an algorithm that can help us identify the location of roots even more quickly.
Newton's Method requires that you start "sufficiently close" (a somewhat arbitrary specification that varies from problem to problem) to the actual root in order to estimate it with accuracy. If you start too far from the root, then the algorithm can be led awry in certain situations.