A thing that does best in {its worst case}.
See minimax_and_bayes_estimator.pdf for a formal description of a minimax decision rule (in the context of Parametric statistical inference as an example), which underlies the general concept of minimax, as found in several areas of applications
Theorem 2. Let {f θ : θ ∈ Θ} be a family of PDFs (PMFs), and suppose that an estimator δ ∗ of θ is a Bayes estimator corresponding to an a priori distribution π on Θ. If the risk function R(θ, δ ∗ ) is constant on Θ, then δ ∗ is a minimax estimator for θ.
From page 411, Rohagi, Sale, "An introduction to probability and statistics"
Note that it's not hard to show that the Bayes risk is a lower bound of the minimax risk (which makes sense, as the worst case has to be worse than the average case..., and this is preserved after taking mins)