In the story, Robert Joyner was the man who had killed himself by the tree in the park; Margo and Quentin were the ones to discover his body there when they were only nine years old, playing together on their bikes.
We don't know a lot about this man at first. Margo discovers that he was thirty-six years old, a lawyer, and very depressed about the fact that he was going through a divorce.
Quentin ponders this man's situation and often thinks of it, even when he's nearly an adult. Considering the plot of the novel, you might consider Robert Joyner the man who unwittingly shapes the story: Margo and Quentin seem to simultaneously cement their connection to each other with the shared experience of discovering Joyner's body even as they become less close after that event. And when nine-year-old Margo considers that perhaps Joyner's "strings" had come detached, the idea seems to shape her thinking throughout the rest of the novel as she continues to think of humans in terms of having strings. The fact that the teenagers ceremoniously say goodbye to Joyner in the end of the story also suggests his importance to their thinking and to their relationship with each other.