Vui is an Eevee that was experimented on by Team Rocket, thereby gaining the ability to freely change from an Eevee into any of its three Generation I evolutions (Vaporeon, Jolteon and Flareon) and back.
Vui first appeared in Blame it on Eevee. When Red tried to challenge Erika, she stated that she wouldn't battle him unless he was able to capture a Pokémon of her choice, an Eevee. Red, with the help of Bill, tried to battle Vui, but for every Pokémon that was sent out Vui would turn into the eeveelution that had a type advantage over it. This ability confused both of them at first, but Bill eventually figured out the ability, and Red managed to discover a way to keep Vui from sensing the type of Pokémon sent out. Red was thus able to capture the Eevee.
Vui was later used alongside Red's other Pokémon to battle Team Rocket members who were pursuing Articuno in the Seafoam Islands. Soon afterwards, Red sent Vui into storage after restoring Aero in Cinnabar Island. It was later discovered that when Red had sent him to Professor Oak, Vui had been captured by Team Rocket. Later on, Red rescued Vui in The Winged Legends. Red also used Vui to battle Bruno on Cerise Island, taking advantage of his ability to change into his evolutions, which was enhanced since Red had three of the Vermilion Harbor evolutionary stones.
Vui was used on Red's test to become Viridian City's Gym Leader, defeating the last of the Association's Pokémon, Porygon2. It was his first appearance as an Espeon, and although he was not identified as Vui, Red expressed that he clearly had much trust in the Pokémon. Red later explained to Sabrina that the Pokémon was in fact Vui but that he had lost his ability to change into his Fire, Water, and Template:Type2 evolutions upon evolving into Espeon.
Moves used recently are in bold unless all moves all fit this case.
Trivia
The game variation of Red that appears in the Generation II games owns a level 73 male Espeon, which most speculate is the representation of the Eevee that the player receives in Celadon City, just as Red caught Vui there. However, it is not present in the Generation IV games, instead replaced with a Lapras.