Correlated equilibrium concepts are often related to communication, most easily implemented via a mediator. Communication can archive two things. It helps with coordination and it can transmit private information. In games of complete information, there is no private information and only the first part matters.
Let's start with the case of complete information:
Ann and Bob play a simultaneous move game. They can choose their actions at random. If their random action choices are mutually best responses, they are playing a mixed strategy Nash equilibrium.
Now, Ann and Bob hire a mediator, Claire, to help them coordinate. Claire randomly selects what both Ann and Bob should do, tells Ann what she should do, and tells Bob what he should do. If Ann and Bob are always willing to follow Claire's recommendations, we have a correlated equilibrium. Since Claire could mimic individual randomizations, every mixed strategy Nash equilibrium can be implemented by Claire and is, therefore, a correlated equilibrium.
Let's add incomplete information:
Ann and Bob have now private information, but Claire does not know anything Ann or Bob does not. Now Claire asks Ann and Bob for their private information and only then makes a recommendation as before. If both Anna and Bob are always willing to tell Claire their private information and are also willing to follow her recommendation, we have a communication equilibrium.
Ann and Bob have now private information and so does Claire. In particular, Claire can make recommendations that use private information neither Ann nor Bob have. Now Claire asks Ann and Bob for their private information and only then makes a recommendation as before. If both Anna and Bob are always willing to tell Claire their private information and are also willing to follow her recommendation, we have a Bayes correlated equilibrium.
Apart from Bayes correlated equilibria, you can find a nice discussion of these concepts in Myerson's graduate game theory textbook "Game Theory: Analysis of Conflict" in Chapter 6.