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In addition to long-existing load uncertainty on power systems, continuously increasing renewable energy injections (such as wind and solar) have further made the power grid more volatile and uncertain. Stochastic and recently introduced robust optimization approaches have been studied to provide the day-ahead unit commitment decision with the consideration of real-time load and supply uncertainties. In this paper, we introduce an innovative minimax regret unit commitment model aiming to minimize the maximum regret of the day-ahead decision from the actual realization of the uncertain real-time wind power generation. Our approach will ensure the robustness of the unit commitment decision considering the inherent uncertainty in wind generation. Meanwhile, our approach will provide a system operator a clear picture in terms of the maximum regret value among all possible scenarios. A Benders' decomposition algorithm is developed to solve the problem. Finally, our extensive case studies compare the performances of three different approaches (robust optimization, minimax regret, and stochastic optimization) and verify the effectiveness of our proposed algorithm.