The two-parameter Weibull distribution is the most widely used distribution in reliability engineering. Parameter estimation is a key issue to apply the Weibull to practical engineering. This paper aims to compare the performance of the maximum parameter estimation method (MLE) and the moment parameter method. It firstly investigates some mathematical properties such as solution uniqueness, estimator's equivariance and the confidence interval, that are important to practice. Later on their performances have been evaluated by using simulation. In the simulation, data sets ranged from extreme small to large have been considered. Weibull distribution with increasing, constant and deceasing failure rate have been chosen in the simulation to ensure the simulation's results concrete.