In F. Scott Fitzgerald's The Great Gatsby, Chapter Eights provides Nick Carraway (the narrator) with Jay Gatsby's history with Daisy.
Gatsby tells how he ended up in her beautiful house all those years before when he was in the military service. It was a place she didn't really seem to notice...it was the way she had always lived. However, Jay was great impressed by the home, the life that exuded from it, and most especially by Daisy herself:
He went to her house first with other officers from Camp Taylor, then alone...But what gave it an air of breathless intensity was that Daisy lived there—it was as casual a thing to her as his tent out at the camp was to him.
The house speaks of life coming from a variety of directions. For instance, Daisy is pursued by many men, and this also draws Gatsby—the pursuit of others makes her seem all the more valuable to him somehow. However, what makes his presence a "colossal mistake" is Jay's background. Jay's parents were lazy farmers. As soon as he had a chance, young James Gatz had run away, forgetting how his parents were; he decided not just to change his name, but also his "identity"—to someone he wanted to be. Gatsby tells Carraway...
However glorious might be his future as Jay Gatsby, he was at present a penniless young man without a past, and at any moment the invisible cloak of his uniform might slip from his shoulders. So he made the most of his time. He took what he could get, ravenously and unscrupulously—eventually he took Daisy one still October night, took her because he had no real right to touch her hand.
Gatsby can honestly see himself as someone who entered Daisy's home—and her life—under "false pretenses. The reason for this is because he is not what he appears to be. He is not the Jay Gatsby he will one day become. It may be simply be an opportunity of chance that is presented to him in a magical moment. He does take advantage of the moment...his life is a lie—he is not who he pretends he is. But Daisy obviously is taken enough with him that she cannot let him go, even after she marries Tom Buchanan.
The concept of a mistake may not just speak to that moment in their past, but may also speak to how that chance meeting will eventually draw Daisy and Gatsby together again...and now she has killed Myrtle Wilson with Jay's car. This would never have happened if he had not been at Daisy's home all those years before.