When your sword becomes moderately or severely damaged, it has a chance to enter an Awakened state (真剣必殺), which is signaled by a cut-in screen with a unique CG sprite. The probability of entering an Awakened state is governed by the sword's Certain Kill (必殺) stat. Afterwards, your sword moves forward and can now attack twice in one turn, and will always deal critical hits.
An Awakened sword will always launch its first counterattack towards the enemy that triggered the awakening. Naginatas and Oodachis that awaken will ignore their usual extended ranges and will only attack the one enemy that triggered their awakening during the counterattack. After the battle, the awakening CG sprite will be saved to your Sword Library.
Awakened state will only last for the duration of the battle, and does not carry over to subsequent battles (unless triggered again). More than one sword can trigger Awakening within the same battle.