Holden's nose is bleeding, because his roommate, Stradlater, punched Holden in the nose.
Their fight occurred in chapter six. Stradlater had just gotten back from his date with Jane. Holden is genuinely concerned about it, because he doesn't want to find out that Stradlater had sex with Jane. Unfortunately, Holden's curiosity gets the better of him, and Holden flat out asks Stradlater what happened. To Stradlater's credit, he doesn't kiss and tell.
"What'd you do?" I said. "Give her the time in Ed Banky's goddam car?" My voice was shaking something awful.
"What a thing to say. Want me to wash your mouth out with soap?"
"That's a professional secret, buddy."
Holden loses it at that point. The lack of specific details angers Holden and Holden start berating Stradlater and calling him a "moron." Stradlater pins Holden to the floor and immobilizes Holden. It's clear that Stradlater doesn't want to hurt Holden, so he just holds Holden down until Holden says that he's done. Holden lies. He gets up and calls Stradlater a moron one more time. Stradlater lets go with a punch that levels Holden and creates a big nose bleed.
That got him really mad. He shook his big stupid finger in my face. "Holden, God damn it, I'm warning you, now. For the last time. If you don't keep your yap shut, I'm gonna--"
"Why should I?" I said--I was practically yelling. "That's just the trouble with all you morons. You never want to discuss anything. That's the way you can always tell a moron. They never want to discuss anything intellig--"
Then he really let one go at me, and the next thing I knew I was on the goddam floor again. I don't remember if he knocked me out or not, but I don't think so. It's pretty hard to knock a guy out, except in the goddam movies. But my nose was bleeding all over the place.