Blue

Blue Oak was the grandson of Professor Oak, and brother of Daisy Oak. He was also the rival to Red, who lived next door to him. After Red picked Charmander, Blue countered by choosing Squirtle.

Gen 1 (Pokémon Red)
Despite the fact that Blue was considered to be the strongest trainer in the game, becoming the Champion of Kanto, he posed little threat to Red, and lost almost all of his fights against him, sometimes in a matter of minutes. The only times Blue won were when Red's team was weakened from previous battles. Most notable was his victory on Route 22, when Red purposely let him win so he didn't have to deal with the ledge in front of the Pokécenter in Viridian City. Blue also won at the time of Red's first battle against him at the Pokémon League, but it was a short moment after ATV's incredible victory over Lance, which let the Bug/Poison-Type Pokémon as the only one still able to fight.

It is not clear which role Blue had in TPP Red, as Red himself was seen more as a carrier of holy and/or powerful beings like Lord Helix, Bird Jesus or King Fonz. Some early lore tried to place Blue as the chosen one of the Dome Fossil, to contrast Red's title of chosen one of the Helix Fossil. However, Blue never had the representation of the Dome, Kabutops, in his party, and it was later theorized that the one working for the Dome was actually Bill, thus Blue became no more than just an annoying rival who stood in Red's way.

Gen 2 (Pokémon Crystal)
Blue returned as the Gym Leader of Viridian City. Like in Gen 1, he did not have much importance to the lore, because of AJ and his Godslayers' goal to defeat an even greater enemy than him: Red and the Chosen Six.

After a very close match with AJ, Blue was defeated and gave him the Earth Badge, granting the boy his sixteenth and final badge. In lore, this allowed AJ to go to Mt Silver and finally meet Red in order for LazorGator to exert his revenge on Lord Helix.

Gen 4.5 (Randomized HeartGold)
Blue is still the Gym Leader of Viridian City. He was challenged by the current host of The Voices, Aurora Lovejoy. After a close match, Aurora takes the win and receives the Earth Badge, going on to fight Kanto's Champion at the top of Mt. Silver, Alice.

Gen 5.5 (Pokémon BlazeBlack 2)
Blue travels to Unova to participate in the Pokémon World Tournament.

Trivia

 * Blue had only one Pokémon in common with Red: a Pidgeot. The stream gave it the nickname of 'Bird Judas' to mirror Red's Bird Jesus. Before the lore considered Bill to be the Dome Fossil carrier, many believed the presence of the bird Pokémon was the mark of Blue's alliance with the God of Democracy.
 * In Gen 4.5 (Randomized HeartGold), Blue's name appeared as Aoooo was about to fight him at Viridian City. It is debated whether it was really Red's former rival or Alice's rival, Green, who changed his name and Pokémon after being defeated by the young girl, out of shame.