Corrected entry: In the scene where Nash and his friends are in the bar discussing how best to approach the blonde and the other girls, Nash suddenly has an idea which later leads to the game theoretic concept of "Nash equilibrium". However, Nash's description of this concept in the bar scene is totally, utterly wrong.
Correction: I don't believe in the movie it is stated he is talking about Nash's equilibrium (as a whole). He is explaining game theory (players' chances improve by considering the best strategy for themselves and the other players). This breakthrough leads him down the path to his equilibrium theory (we see in the movie he spends months working on it after the bar scene).
Correction: How do we know that? Until you can actually say HOW this is "totally, utterly wrong" it can't really be seen as a mistake, because for all we know, it is totally, utterly right.