An algorithm is presented for maximum likelihood estimation (MLE) of three-parameter Weibull distribution in this paper, which is not only suitable for uncensored data but also appropriate for various censored data. Firstly, a unified likelihood equation under censored data and uncensored data was built. Here, the second-order convergent Newton-Raphson iteration method was applied to solve the MLE of two-parameter Weibull distribution, which had only scale parameter and shape parameter with a given positional parameter. Then, the steady and rapid Brent search method was applied to solve the optimal solution of the likelihood function just with a single variable of positional parameter. Finally, several examples were given to demonstrate the stability and efficiency of this algorithm.