What is the basic difference between Nash equilibrium and Bayesian equilibrium?
Roughly speaking, Bayesian Equilibrium is an extension of Nash Equilibrium for games of incomplete information. In these types of games, players do not know the state of nature (but know the set of possible states of nature). The Bayesian approach is most useful in dynamic games (Perfect Bayesian Equilibrium).
For example, if you are playing poker with people you don't know, you are uncertain about the other players' risk preferences: they could either shy away from risk or be more daring. You start out with some prior about their risk aversion and choose your strategy so as to maximize your expected payoff given that prior. As the game progresses, you will gather evidence from the hands played and update your belief about each player's risk preference accordingly.
A Perfect Bayesian Equilibrium puts all these elements together: you have some prior belief (probability distribution) about the opponent's type, which you update according to Bayes' rule as the game progresses; and a strategy that is consistent with this belief. Finally, as in Subgame Perfect Equilibrium, PBE imposes that optimal play and Bayesian updating happens in every subgame (that is reached with positive probability).