Aragorn and Legolas are fighting because Aragorn insulted Legolas hair.
Legolas: What's wrong with my awesome and fabulous hair!?
Aragorn: Just like Vivi once said... You look like a girl.
Legolas: Every elf has long hair!
Aragorn: Then you're all girls!!!
Legolas: NOW YOU'RE GOINH TO FAR! *Tries to hit him*
Aragorn: AHAHAHA YOU HIT LIKE A GIRL!!!
Legolas: *Punches Aragorn in the face*
Aragorn: That's it! But you still hit like a girl!
Legolas: I HATE YOU!!!! *Cries like a baby*
Frodo: Whoah! What happened here?
Aragorn: Frodo? What are you doing here?
Frodo: Just walking... Why is Legolas crying?
Aragorn: Because I insulted him...
Frodo: Aha.
Aragorn: I got to go now... *Walks away slowly*
Frodo: Yeah, me too... *Walks away akwardly*