The Pokémon League entrance exam instructor is a minor character from the episode "The Ultimate Test." His name was never officially revealed.
He oversees the Pokémon League Admissions Exam and handles the final battle portion, where he loans test-takers random sets of three Pokémon to battle him with. When Ash, Team Rocket, and Nurse Joy all signed up for the exam, the instructor collected their Pokémon at the counter (though Ash handed Pikachu to Misty since he couldn't use his own team during the test). He proctored the written portions of the exam, and when Jessie started getting mouthy about the difficulty, he simply expelled her.
The battle rounds were where things got chaotic. Nurse Joy faced off against another instructor and won, then James took his turn—but immediately got disqualified for trying to send out two Pokémon at once. Ash's match came next, and he was pitted against the instructor's three Eeveelutions. During the fight, Meowth freed a Meowth that had been frozen by Vaporeon, and the instructor wanted to catch it. Before he could, James returned with the instructor's loaned Pokémon and ordered them to attack their master. The instructor simply commanded them to turn on Team Rocket instead, which they did, helping to finish them off.
After Team Rocket was dealt with, the instructor offered to let Ash retake the exam since it had been disrupted. Ash turned him down, preferring to pursue entry to the Pokémon League through actual battles instead.
Content compiled by AnimeList.moe from publicly available sources.