The crime of assault was the act of making physical contact with someone in order to injure or incapacitate them.
While discussing options to end the hostage situation at the monastery at P'Jem in 2151, Sub-commander T'Pol mentioned that an armed assault would put all lives at risk, not to mention the destruction of the sanctuary itself. (ENT: "The Andorian Incident")
Striking a fellow officer was a court martial offense in the Federation Starfleet. Both Captain James Kirk and Mr. Spock were guilty of this offense in 2267, but, since they were the only two crewmembers still left aboard the USS Enterprise, Spock agreed with Kirk's logic that if they were both confined to the brig, no one would be available to build the subsonic transmitter needed to destroy the influence of the Omicron spores and return the crew to the ship. (TOS: "This Side of Paradise")
Hikaru Sulu and James Kirk each assaulted officers of Federation Security while removing Leonard McCoy from a prison. They were charged by the Federation Council with those acts shortly afterward, but those were summarily dismissed by the President of the Federation. (Star Trek III: The Search for Spock; Star Trek IV: The Voyage Home)
In some jurisdictions, such as Bajoran territory, assault could also include directing weapons fire at another individual. Lissepian Mother's Day Heist conspirators Hain, Nahsk, Krit, and Larell were all arrested for assault with intent to kill in 2374 after they all discharged weapons at each other in attempts to claim the proceeds of the Heist. Odo assured Quark, who was also put in danger, that they would go to prison for "long enough". (DS9: "Who Mourns for Morn?")
After seven USS Voyager crewmembers, including Tom Paris and Harry Kim, got into a brawl with 14 individuals aboard the Markonian outpost in 2376, all twenty-one were arrested with the charges ranging from disorderly conduct to assault on a security officer. (VOY: "Survival Instinct")