After the accident in which Daisy hits and kills Myrtle, Gatsby opens up to Nick. He tells him about becoming an army officer in World War I and meeting Daisy. For the first time ever, Gatsby is invited into the house and life of a wealthy person like Daisy as an equal.
The first misunderstanding is Gatsby's misleading Daisy into thinking he is her social equal, with a monied family behind him, not a poor boy from nowhere. More profoundly, he misunderstands himself in thinking he can take what he can get from the relationship with Daisy and walk away unscathed. Instead, he falls in love. As Nick puts it:
now he found that he had committed himself to the following of a grail .... He felt married to her.
While Daisy reciprocates Gatsby's love for a time, his deepest misunderstanding is that she will feel "married" to him and be as unwaveringly constant as he is to her. He doesn't understand that her love for him is much more shallow than his for her.
This misunderstanding repeats itself at the end of Gatsby's life. Once again, they have an affair. Once again, for Gatsby, it is filled with cosmic significance, and he expects Daisy to leave Tom and go off with him. However, Daisy treats her time with Gatsby as more of a fling, a passing interlude in her life. Gatsby waits again for Daisy, as he did during the war—this time with tragic results.