Blue is the grandson of the great Professor Oak. In the Kanto based games, he is the player (Red, Green, or Leaf's) rival. He becomes the Kanto champion as well until you defeat him. In the Johto based games and Let's Go, he has become the Viridian City Gym Leader after being defeated by Red and taking Giovanni's place. In the Unova based games, he's still the Gym Leader of Viridian City, but participates as a former Champion. In the Alola based games, he's considered a legend upon legends, and is the oldest he's ever been.
Blue is Red's rival in Pokemon Origins and Generations. He is also what Gary Oak is based off of.