Who are the two childhood friends of Hamlet and what is the significance of them appearing in the play?

Rosencrantz and Guildenstern are Hamlet’s childhood friends, and he has them killed when they betray him at Claudius's behest.

Rosencrantz and Guildenstern are Hamlet’s school friends. They return at Claudius’s request, and Claudius tries to manipulate them. He tells them that Hamlet has gone crazy because of his father’s death.

Welcome, dear Rosencrantz and Guildenstern!
Moreover that we much did long to see you,
The need we have to use you did provoke
Our hasty sending. Something have you heard
Of Hamlet's transformation; so call it,
Sith nor the exterior nor the inward man
Resembles that it was. (Act 2, Scene 2) 

Hamlet, for his part, views his friends’ presence as an interesting development. He seems to be aware of what his uncle is doing. He wants to make sure that they are aware that his crazy act is just an act. He tells Guildenstern that he is not crazy. 

I am but mad north-north-west: when the wind is
southerly I know a hawk from a handsaw. (Act 2, Scene 2) 

Hamlet is definitely not crazy. It is interesting that he wants these two to know it, because he doesn’t let many others in on the secret. Apparently he doesn’t want people who used to respect him to think that he went off the deep end.

Later, Hamlet tells Rosencrantz that he is a “sponge,” meaning that he is being manipulated by Claudius. Hamlet’s point is that he knows what Rosencrantz and Guildenstern are up to. He can play the game better than they can. When Claudius sends Rosencrantz and Guildenstern with a letter that is supposed to result in Hamlet’s death, Hamlet turns the tables on them and writes his own letter so that they are the ones killed instead. He is not about to be betrayed by either his uncle or his childhood friends.