In chapter 2 of Ender's Game, why does peter flop onto his bed and laugh?
The primary purpose of the fight scene between Ender and his older brother Peter in chapter two, is to reveal the character of Peter, and to show the stark contrast between his character and Ender's character. Peter is, at his core, evil and power-hungry, and therefore manipulative.
When fighting with Ender in a game called "Buggers and Astronauts," we see Peter bullying his baby brother like any big brother would do. But Peter seems to take things further than what is appropriate. He pushes things as far as they should go, then pushes them further. After pushing Ender almost to the point of tears (for physical pain and emotional pain), he then flops back on his bed laughing. In the immediate context of the story, this moment suggests that Peter is just joking and that we aren't to take him too seriously. But this actually ironic.
In the context of the story as a whole, he does this to make Ender (and Valentine) afraid of him. He does it to keep his siblings guessing about him, making sure there is nothing about him that is predictable. He also very likely uses his siblings as "practice" to see exactly what he is capable of in coercing people to do what he wants. It is impossible to tell what Peter's motives are, and it is impossible to tell when he is acting and when he is being serious. This is exactly the point. Later in the story, when Peter again masks himself on the Nets as "Locke," we are left knowing one thing: he wants power and will do anything to get it. It is impossible to know if Peter has the capacity for love, but the assumption is that he does not.