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There is a natural one-to-one correspondence between squarefree monomialideals and finite simple hypergraphs via the cover ideal construction. Let H bea finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomialring R. We give an explicit description of all associated primes of R/J^s, forany power J^s of J, in terms of the coloring properties of hypergraphs arisingfrom H. We also give an algebraic method for determining the chromatic numberof H, proving that it is equivalent to a monomial ideal membership probleminvolving powers of J. Our work yields two new purely algebraiccharacterizations of perfect graphs, independent of the Strong Perfect GraphTheorem; the first characterization is in terms of the sets Ass(R/J^s), whilethe second characterization is in terms of the saturated chain condition forassociated primes.