It is actually only in Chapter Three that Nick meets Gatsby for the first time. This momentous event occurs at one of Gatsby's party, where Nick mentions that he was one of the few people who actually received an invitation to go. Nick is sitting at a table with Jordan Baker and another man who is with "a rowdy little girl". During a brief quiet moment, the man begins to start a conversation with Nick and they discuss their similar backgrounds, focussing on their involvement in the war and their experiences in France. The man then tells Nick about the hydroplane he has just bought and asks if Nick would like to try it out with him the next day. It is only when Nick reveals to his new friend that he has never met Gatsby that the man reveals himself to be Gatsby:
"This is an unusual party for me. I haven't even seen the host. I live over there -" I waved my hand at the invisible hedge in the distance, "and this man Gatsby sent over his chauffeur with an invitation."
For a moment he looked at me as if he failed to understand.
"I'm Gatsby," he said suddenly.
It is in this way that Nick meets and discovers the identity of Gatsby in Chatper 3.
When Nick first sees Gatsby, it is during the last two pages of chapter 1. The two never talk in this scene. Nick is standing in his own yard overlooking the vast landscape of the mansion next door. As he watches he notices a man out in the evening in Gatsby's yard. Nick confirms for us this is indeed Gatsby, but the interesting point of this scene is what Gatsby is doing. Nick says his arms are reaching out with longing. He is facing the water and Nick follows this man's gaze and reach with his eyes. Nick can see him reaching for a green light at the end of a dock in the distance.
This scene is important because later, we learn who the dock belongs to and why Gatsby would long for what it represents. As Nick watches this green light, when he turns back to find Gatsby, the image of Gatsby is gone. Nick actually uses the term "vanished".