The Newton-Raphson method is a numerical technique used to find approximate solutions to equations, particularly for finding roots. It starts with an initial guess and iteratively improves this guess using the function and its derivative. The formula involves taking the current guess, subtracting the function value divided by its derivative, and repeating this process until the solution converges to a desired level of accuracy. This method is particularly effective for functions that are continuous and differentiable. However, it may fail to converge if the initial guess is too far from the actual root or if the derivative is zero at the guess.