A pursuer is someone who is chasing someone or something. You could be a pursuer of truth and beauty, or perhaps a pursuer of the kid who took your lunch money.
To pursue is to chase something, like a dog pursuing a squirrel. In that situation, the dog is a pursuer. Someone who’s trying to ask someone else out on a date is a pursuer. When the police are on a manhunt for a criminal, the police are pursuers. You could also say someone is a pursuer of a goal, like a pursuer of a medical degree. All pursuers want something and they're going after it.