This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut +△ (ε△μ-ε-1f(u))=0. It is shown that the a posteriori error bounds depends on ε-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm for computing the solution of the Cahn-Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.