Norman runs the Petalburg Gym and hands out the Balance Badge to trainers who beat him. He's all about Normal-type Pokémon—hence the name—and knows his type inside and out.
His gym is basically a maze of branching hallways, each door labeled with a hint about what trainer waits inside: someone who spams X Attack, someone who chugs Potions constantly, that sort of thing. Eventually you reach Norman himself. In Ruby and Sapphire, he fields two Slaking and a Vigoroth; Emerald switches things up with a Zangoose and Spinda instead. Win and he gives you TM42: Facade, which deals double damage if you're hit with a status condition.
In Pokémon Adventures, Norman is Ruby's father and his character stays pretty faithful to the games—calm and collected. He was originally set to become a Hoenn gym leader five years before the main story, and he and Professor Birch took their families to scout the region. That's when a wild Salamence attacked Sapphire, and Ruby jumped in to protect her, earning a scar on his forehead in the process. The incident scared the Salamence into a nearby research lab, which accidentally released Rayquaza. As punishment, Norman got banned from becoming a gym leader until he tracks down where Rayquaza ended up. The whole thing left its mark on both kids—Ruby decided he needed to be gentler, while Sapphire realized she had to toughen up.
In the anime, he's May and Max's dad and basically a local legend. People say his strength rivals an Elite Four member's. His signature Pokémon are a high-level Slaking and a powerful Vigoroth. His undefeated record finally breaks when Ash challenges him for his fifth badge and wins. Max doesn't handle the loss well—he's so upset seeing his dad lose that he grabs the badge right out of Norman's hands and bolts. Ash, May, and Norman eventually track him down and help him understand that losing is just as important as winning. Max comes around and returns the badge.
Content compiled by AnimeList.moe from publicly available sources.