Aang entered the Avatar State on multiple occasions during Avatar: The Last Airbender, and every time it happened, Aang demonstrated what made his powers so unique. Once Aang returned after a 100-year hibernation, the last living Airbender didn't have any time to waste in order to save the world. Aang had to quickly master the arts of Waterbending, Earthbending, and Firebending so that he would be powerful enough to defeat the Fire Nation. With each new bending ability he learned, Aang gained more powers, but being the Avatar gave him another special ability.
All Avatars have the power to enter into what is known as the Avatar State. The Avatar State gives the Avatar access to cosmic energy and grants them a connection to past Avatars, which can increase their bending capabilities. In some instances, the use of the Avatar State allows the Avatar to take on the appearance of previous iterations. The power that comes with the Avatar State can be dangerous though, as only Avatars who master the state of mind can control it. An Avatar's eyes glow (as do the tattoos of Airbenders) whenever they are in the Avatar State, but those who have mastered the use of it only see their eyes glow at the moment they enter the Avatar State instead continuously. Avatars might also enter the Avatar State on accident if they are in danger or in an extremely emotional place.
Throughout the three seasons of Avatar: The Last Airbender, Aang is shown using the Avatar State several times and for a variety of reasons. He's entered the Avatar State reflexively to save his life and out of anger or sadness, but Aang also purposefully entered the Avatar State. Aang eventually learned to master the Avatar State too and was shown using it again during The Legend of Korra. The following is every time Aang entered the Avatar State and what happened.
Returning After 100 Years
Aang is first shown using the Avatar State in the debut episode of Avatar: The Last Airbender. He emerges from an iceberg after 100 years (an experience that contributed to Aang's early death in The Legend of Korra). It is revealed later on in the show that Aang only entered the Avatar State after crashing into the ocean during a storm. The Avatar State was activated to save him and Appa from dying and left them in suspended animation for 100 years until Katara and Sokka found them. Katara cracked the iceberg open, which allowed Aang to end his time in the Avatar State.
At The Southern Air Temple
The next time Aang enters the Avatar State is in season 1, episode 3, when he visits the Southern Air Temple with Katara and Sokka. It is here that he learns Firebenders killed his former teacher, Monk Gyatso, while Aang was in the iceberg. Aang's emotions cause him to enter the Avatar State again, but Katara manages to calm him down with the reassurance that Aang isn't alone anymore.
At The Fire Temple
Aang entered the Avatar State again during season 1, episode 8. For his third time in the Avatar State in The Last Airbender, the show added a new twist to what Aang could do. He took on the form of Avatar Roku during a visit to the Fire Temple, which had a sanctuary that allowed Aang to connect to the former Avatar. Roku used his firebending abilities to save Katara and Sokka and destroy the temple.
During The Storm
Aang is next shown entering the Avatar State during a large storm in season 1, episode 12. The storm comes while Sokka is out on the sea with fishermen and all of their lives are put in danger. Aang and Katara fly through the storm on Appa's back to save them and do rescue Sokka and the fishermen. Unfortunately, the storm sends them all crashing into the ocean. Aang re-enters the Avatar State here to use Airbending to create a protective sphere around them so they don't drown.
At The Northern Water City
Season 1, episode 20 of Avatar: The Last Airbender featured the greatest showcase of Aang's ability up to that point when using the Avatar State. It happens when the Fire Nation are invading the Northern Water Tribe and after Admiral Zhao kills the Moon Spirit's mortal koi fish form. Aang then enters the Avatar State and combines himself with the Ocean Spirit, forming a giant koi fish made of water. Aang uses his increased power to wipe out the Fire Nation army and save the Northern Water Tribe. His connection to the Ocean Spirit ends once the Moon Spirit is restored by Yue.
To Save Katara
The first episode of season 2 of Avatar: The Last Airbender shows Aang entering the Avatar State again to save Katara. General Fong of the Earth Kingdom pushes Aang to learn to master the state and uses Katara as bait to trigger the Avatar State. Fong tricks Aang into believing he killed Katara and Aang's emotions activate the Avatar State again. Aang can't control it, even after Katara is confirmed to be alive, and destroys his surroundings.
After Appa's Disappearance
Aang enters the Avatar State again in season 2, episode 11 when he confronts the thieves who stole Appa. The Earthbenders kidnapped the flying bison and sold him, but do not know where Aang can find him. This infuriates Aang and he enters the Avatar State again. Katara manages to calm Aang down before he hurts anyone, although he does destroy the thieves' mode of transportation.
Fighting Zuko & Azula
The season 2 finale of Avatar: The Last Airbender sees Aang enter the Avatar State again. After not being able to let go of his attachment to Katara in exchange for cosmic power, Aang believed he wouldn't be able to master the Avatar State. But, when a fight with Zuko and Azula turns dire, Aang decides to let go of Katara so he can save them all. Aang enters the Avatar State ready to defeat the Fire Nation, but Azula hits him in the back with her lightning. Katara manages to save his life by using spirit water to heal him, but Aang remained unconscious for weeks anyway.
Fighting Fire Lord Ozai
Aang didn't enter the Avatar State at any point in season 3 except for Avatar: The Last Airbender's series finale. The fourth and final part of the Sozin's Comet arc saw Aang finally confront Fire Lord Ozai, who was more powerful than ever before thanks to the comet. Aang only enters the Avatar State near the end of the battle and the use of this power allows Aang to overpower Ozai. But, he doesn't want to kill Ozai and exits the Avatar State before delivering a deadly strike. Aang instead decides to use energybending to strip Ozai of his bending abilities.
Fighting Yakone
The last time Aang is known to have entered the Avatar State comes in season 1, episode 9 of The Legend of Korra. Aang is shown fighting a bloodbender named Yakone, who initially gets the best of the Avatar. Aang enters the Avatar State once more to chase down Yakone, and the power kicks in again when the bloodbender attempts to snap his neck. The Avatar State saved Aang's life and allowed him to take Yakone's bending abilities away.