Garth's tactics and actions in this battle were required reading at Starfleet Academy. It was noted by James T. Kirk that Garth's victory was instrumental in making it possible for him and Spock to work together as brothers. (TOS: "Whom Gods Destroy")
Precisely whom Garth defeated at the Battle of Axanar has not been established in canon. There has been some speculation that it may have been the Romulans, though, judging by comments made in TOS: "Balance of Terror", it seems there was never any Federation contact with the Romulans during the time frame of the battle. (Star Trek Encyclopedia (4th ed., vol. 1, p. 54))
The Star Trek Encyclopedia (4th ed., vol. 1, p. 54) assumed that, based on Kirk's comments, "the Axanar battle apparently had something to do with holding the Federation together."
The Battle of Axanar was, originally, to have been referenced in DS9: "Valiant". In the first draft script of that episode, the incident was spoken about by Nog while he was in a runabout with Kira Nerys. In the ultimately discarded dialogue, he commented, "…but Captain Garth knew that the enemy shields would be affected by the heavy [TECH] radiation in the Axanar System and his shields would not. In the end, the Federation only lost four ships while the Klingons lost their entire fleet. It's one of the greatest victories in Federation history [....] The Battle of Axanar is… classic. It's the textbook example of how to use a spherical deployment to defeat a superior enemy force." However, Kira argued that the battle, having taken place more than two centuries earlier, wasn't relevant any more. She exemplified this by pointing out to Nog that, in the last two hundred years, literally no-one had replicated the tactic of using a spherical deployment to overcome a superior enemy force.
According to the FASA sourcebook The Four Years War, the battle was fought between a Federation fleet led by Garth of Izar, commanding the USS Xenophon, and a Klingon invasion force led by Admiral Kkorhetza zantai-Riskai. It became the start of a conflict known as the Four Years War.
According to the book Star Trek: The Original Series - Core Game Book, the Axanar Rebellion was a conflict between the Federation and the Axanari. The Axanari were offended by the suggestion that their application for Federation membership should be rejected because of their inferior starships. They responded by conquering several planets and presenting them to the Federation. The Federation demanded that the Axanari immediately withdrew from their conquered planets, which angered them even more. Garth of Izar commanded the USS Constitution during the conflict, and the Federation was able to win despite the Axanari fleet being numerically superior.
According to the book Star Trek: Federation - The First 150 Years, the battle was fought between the USS Constitution, commanded by Garth of Izar, and three Klingon warships.
According to the graphic novel Star Trek: Boldly Go #10, the version of the Battle of Axanar that occurred in the Kelvin Timeline was fought between the Federation starship USS Heisenberg (NCC-1251), commanded by Captain Garth, and three unnamed Klingon warships. The Heisenberg successfully defeated all three warships using a tactic called the Cochrane Deceleration Maneuver that was previously only theorized and never before successfully used in a battle. The maneuver included a full stop, then an immediate tack to starboard point one-two-zero. Captain Garth used this tactic to evade three Klingon torpedoes, then rotate the Heisenberg 180 degrees to directly face the last surviving Klingon ship, which allowed a clear phaser shot, and the warship was destroyed.