All the action happens off-screen, but reportedly it was the judge throwing punches. His conduct on (and off) the bench is terrible, but it also seems like it was a bad idea for the public defender to apparently take him up on his challenge by storming back there in response to a judicial taunt.
According to this report, it's the judge the audience is applauding at the end of the clip, but neither one deserves anything but shocked silence.
Via Gideon at A Public Defender.
Update: Later reports said that both combatants would be taking some time off while the Florida Bar and (probably) the Judicial Qualifications Commission look into the matter. One report quoted another local attorney who knows both the fighters as saying the incident surprised him, and he wondered if it had resulted from "a misunderstood prank by the judge." You never know. Maybe they are both in a local Shakespeare company and suddenly realized they needed to rehearse a scene from Macbeth. Let's not rush to judgment.