Details: After defeating the Main Storyline, by going to the Thrifty Mega Mart in Royal Avenue, you can speak to Kagetora (cashier) and he will ask you for your help. Apparently long, long ago - thirty years to be exact - he idolized this certain group of trainers, ones who specialized in training a given Eeveelution, and he always tried to defeat each member of the group. But he could never fulfill his goal due to life circumstances, primarily due to getting married, having a child and starting a family.
So, he asks if you could pick up where he left off, to travel across all of Alola to find and defeat each Eeveelution Trainer. He will give vague clues on the location of each of them, but, ultimately, it is up to you to find them for yourself. Each trainer only has one Pokemon when you Battle them, one of the various Eeveelutions. And all but one of them will have their given Eeveelution at Level 55.