Legion Training is the fighting style that the Roman Demigods learn, practice with and use at Camp Jupiter. In combat, Roman legionaries would attempt to outsmart and outlast enemies stamina-wise. Historically, the Roman general that standardized legion training and state-issued equipment was Consul Marius, shortly before the reign of Caesar.
As shown by Jason Grace, it's mostly about quickly learning your opponent's fighting style and then adapting to it. This style of fighting is most effective when in groups. While Jason was fighting Lityerses, he started analyzing patterns and learning his opponent's style which was no defense, only offense, he countered, sidestepped, blocked, and then overwhelmed Lityerses. Jason called it legion training, most likely due to his unique way of fighting and as Lityerses had never even heard of Rome, it was even more effective.
During the war games, Percy's unorthodox, solo, and unpredictable fighting style was dangerous for the campers at Camp Jupiter because they were used to fighting in an organized group. Because of this, Percy was able to defeat multiple enemies in a short period of time, inducing mass panic.
Later, when Percy escapes from the Amazons, he sidestepped, hacked a spear in half, and then hit the warrior with the butt of his sword. Frank pointed out that it wasn't any Roman technique he was using.