After mastering all four elements, the Avatar is usually the strongest being alive in their era, with the Avatar State ensuring that an Avatar could defeat opponents with more years of experience. It allows the Avatar to command the energy and knowledge of their past lives to perform incredible feats of bending, and in many cases, is the Avatar's last resort.
Aang utilized the Avatar State a few times back in Avatar: The Last Airbender, and while Korra initially found it a challenge to connect with her past lives on Legend of Korra, she made the Avatar State one of her most frequent and practical weapons after learning how to tap into such incredible power.
"Endgame" (Book One, Episode 12)
The first time Korra tapped into the Avatar State followed her final battle with Amon. After Aang's spirit restored her bending, she immediately entered the Avatar State, bending each element in a show of force that proved that her bending had been restored.
Seconds later, she was shown accessing the powers of the Avatar State once more in order to restore Lin Beifong's Earthbending.
"Rebel Spirit" (Book Two, Episode 1)
While Aang tried long and hard to master the Avatar State, Korra had already possessed a solid grasp of it by Book Two's start, so much so that she used it just so she could win an Air Scooter race against Tenzin's children.
In the same episode, she used the Avatar State during her fight against a dark spirit. Although she seemed to gain the upper hand for a brief second, even the Avatar State wasn't enough to pacify her ethereal foe.
"The Southern Lights" (Book Two, Episode 2)
When Korra went to open the Southern spirit portal, she found herself attacked by dark spirits that quickly restrained her.
The Avatar State not only gave her the strength she needed to break free from their grasp, but it was also just what enabled her to open the spirit portal with a simple touch.
"Civil Wars: Part 2" (Book Two, Episode 4)
After finding out that Unalaq imprisoned Tonraq and a few other Southern Water Tribe warriors on a Northern Water Tribe ship, Korra entered the Avatar State to bend part of their blockade aside so that Varrick's yacht could approach the vessel where Tonraq and the others being kept on.
Not only was this an incredible show of force, but it showed just how lucky the Southern Water Tribe was to have the Avatar on their side.
"Peacekeepers" (Book Two, Episode 5)
While traveling to the Fire Nation to receive aid for the Southern Water Tribe, Korra found herself attacked by her cousins, Desna and Eska. Their battle came to a brief halt after a dark spirit intruded on the fight, prompting the twins to retreat.
Korra entered the Avatar State and attempted to purify the spirit, but given how new Spiritbending still was to her, she failed and was promptly swallowed whole by the spirit.
"Beginnings: Part 2" (Book Two, Episode 8)
After losing her memories due to succumbing to a dark spirit in "Peacekeepers" the Bhanti of the Fire Nation proceeded to lower Korra into special spirit waters that not only restored her memories but allowed her to see how the first Avatar came to be.
After awakening from her slumber in the spirit waters, Korra briefly entered the Avatar State as she reconnected with Raava and regained her memories.
"The Guide" (Book Two, Episode 9)
In "The Guide", Korra struggled to enter the spirit world, but with Jinora's help, she was able to make the transition. The moment the two successfully crossed over, Korra went into the Avatar State.
"A New Spiritual Age" (Book Two, Episode 10)
When Korra met Unalaq in the spirit world, she found herself at a disadvantage. Without her bending and with Jinora being held hostage, she was at her uncle's whim and was forced to open the Northern spirit portal.
Similar to when she opened the Southern spirit portal, she entered the Avatar State to do so.
"Darkness Falls" (Book Two, Episode 13)
Alone, Korra's strength wasn't enough to subdue Vaatu after he'd been freed from his prison, forcing Korra to tap into the powers of the Avatar State to gain the upper hand. Right before she managed to seal Vaatu away, Unalaq showed up.
After Unalaq succeeded in merging with Vaatu and becoming the world's first Dark Avatar, Korra tapped into the powers of the Avatar State once again in order to put him down, but unfortunately, Unalaq's powers were stronger, enabling him to rip the source of the Avatar State, Raava, right out of her.
"Light In The Dark" (Book Two, Episode 14)
After dealing with UnaVaatu, Korra and Raava reunited, and she entered the Avatar State for a brief second, signifying the return of the Avatar in what would be a new Avatar Cycle.
"A Breath Of Fresh Air" (Book Three, Episode 1)
After Korra left both spirit portals open during Harmonic Convergence, the world began to change, both for better and for worse. Spirit vines began to grow across Republic City, inconveniencing many of its residents. In an effort to clear some of these vines, Korra went into the Avatar State and attempted to Spiritbend them away.
Unfortunately, this only made things worse, showing that the Avatar State wouldn't be the answer to all of Korra's problems.
"Venom Of The Red Lotus" (Book Three, Episode 13)
After successfully abducting her, the Red Lotus bent mercury into Korra's body in order to force her into the Avatar State. They then planned to kill her while she was in this state, ending the Avatar Cycle for good.
While this forced Avatar State wasn't ideal, it definitely aided her in evading the Red Lotus's lethal attacks and allowed her to keep up with Zaheer as he exercised his superior powers of flight during their final confrontation.
"The Calling" (Book Four, Episode 4)
After Korra bent the remaining traces of mercury from her body, she entered the Avatar State for the first time in three long years after struggling to access these powers from the moment Zaheer had been defeated.
"The Battle Of Zaofu" (Book Four, Episode 6)
When Korra and Kuvira fought for control over Zaofu, Kuvira seemed to have the upper hand, but after Korra entered the Avatar State, the fight shifted in her favor.
She would've crushed Kuvira if not for the fact that she still had to get over the mental trauma that she'd been dealing with for the past three years.
"Beyond The Wilds" (Book Four, Episode 9)
After Jinora and several other tourists found their souls trapped in the spirit world after being snatched up by spirit vines in Republic City, Korra was forced to visit Zaheer, who helped her get into the spirit world and overcome her self-doubt.
In order to free the trapped souls, Korra entered the Avatar State to destroy a bubble of energy that surrounded them, mirroring the way she opened the spirit portals.
"Kuvira's Gambit" (Book Four, Episode 11)
After capturing Bataar Jr., Korra entered the Avatar State in an attempt to scare him into telling her how to take down Kuvira's giant mecha suit.
Fortunately for him, he knew that Korra wouldn't harm a hair on his head, especially with his mothers and siblings standing right behind her.
"Day Of The Colossus" (Book Four, Episode 12)
Korra entered the Avatar State for a brief second while working in tandem with the other Airbenders to bombard Kuvira's mecha suit with bursts of wind that nearly knocked it over.
After a brief retreat, she entered the Avatar State once more to distract it, amplifying her Airbending so that she could avoid the spirit energy cannon's destructive beams at the last second, a feat that not even Airbending masters like Jinora and Tenzin were able to accomplish.
"The Last Stand" (Book Four, Episode 13)
Korra used the Avatar State not to defeat Kuvira, but save her life after she bent away a spirit beam from the detached spirit energy cannon.
This, in turn, created a new spirit portal that fans of The Legend of Korra graphic novels know was put to good use by Asami and Korra during their well-deserved vacation in the spirit world.