In the anime
Oswald is a space enthusiast residing in a small town in Kanto, though his technology, which he purchased via comic books, failed to work. When the townspeople were puzzled by several Clefairy stealing all of their items, Oswald proposed the Clefairy were actually aliens who were using the belongings to help build their spaceship. However, nobody believed his theory except for Ash and his friends.
Along with Jigglypuff, Oswald and the others searched for the Fairy Pokémon and discovered they were indeed making a spaceship with the stolen items. Just as the ship was about to take off, Ash and his friends fled. However, Oswald stayed with the Clefairy, wanting to learn more about them as aliens.
Though the Clefairy and their UFO reappeared in Wish Upon a Star Shape, Oswald was not physically seen and only appeared in Brock's flashback.
In the manga
In The Electric Tale of Pikachu manga
Professor Orville (Oswald) in The Electric Tale of Pikachu
Oswald appeared in Clefairy in Space. In VIZ Media's English translation, he has the name Professor Orville. His role in the manga is nearly the same as in the anime, as he is a scientist who believes that the Pokémon Clefairy come from outer space.