He's an unfriendly young man who usually utters the phrase "Smell ya later" once he's about to leave. Like Red, Blue's main goal is to become the greatest Pokémon master and he actually does manage to do this. Red doesn't hear about this until he beats the Elite Four. Once he does he battles Blue and wins, making Blue lose his title of Pokémon champion.
Three years after that, it's revealed that he has become the new Viridian CityGym leader after Giovanni fled the gym three years earlier. What makes him different from other Gym leaders is that he doesn't have a specific type of Pokémon.
Professor Oak is his grandfather and says that he doesn't know how to care for his Pokémon. This is probably inferred when he leaves Silph Co. while Team Rocket is overthrowing it, leaving Red to take them down.