From Bulbapedia, the community-driven Pokémon encyclopedia.
Ho-Oh (Japanese: ホウオウ Houou) is a legendary Pokémon that Ash Ketchum has sighted several times. It has shown up at a few significant moments in his life. It is the first Generation II Pokémon to appear in the anime.
Legends say that Ho-Oh lived in Tin Tower in ancient times. However, war broke out, and when the Brass Tower was burned, Ho-Oh created Raikou, Entei, and Suicune and then left. A new Tin Tower was created, but Ho-Oh never returned.
Ash first saw this legendary bird in Pokémon, I Choose You!. It appeared on his way to Viridian City during his first day as a trainer. He was lying on the ground with his injured Pikachu and saw that it flew over the rainbow that appeared in the sky after the storm. His Pokédex could not identify it and explained that many Pokémon were still unknown. In Pokémon Emergency!, he tried to explain to Professor Oak what he had seen, though the image he used to point it out appeared to be of an Articuno rather than Ho-Oh. Either way, Professor Oak didn't believe him and brushed it off as Ash being mistaken.
The topic was brought up again in A Ghost Of A Chance. While Ash was in Johto, he explained his encounter to Eusine, who also did not believe that Ash had seen this legendary Pokémon. In For Ho-oh the Bells Toll they heard the bells ringing in Tin Tower intended to summon Ho-Oh. At this point, Eusine was so upset by Ash's claim that he challenged him to a battle.
As fate would have it, Ash saw Ho-Oh a second time in Gotta Catch Ya Later! after he parted ways with his close friends Brock and Misty. The third time he saw it was in Battling the Enemy Within after Ash informally challenged the Battle Pyramid.
Fanon speculation has often focused on it and whether its appearance in Ash's life has signified him as a special trainer. There is also belief that it is Ash's father.
For more information on this Pokémon's species, see Ho-Oh.