Within many routes and areas across Alola, you will find certain trainers that won't Battle you until AFTER you defeat all the trainers on that given route or in that given area, often calling themselves Z Trainers. These trainers often used Z-Crystals during battles, as well as give you special items after defeating them.
Z-Trainers
Route 3 - Ace Trainer Makana
Pokemon: Level 13 Rockruff, Level 14 Slowpoke
Reward: Red Card
Diglett Tunnel - Black Belt Greg
Pokemon: Level 23 Hariyama
Reward: Max Revive
Pokemon: Level 22 Slowpoke
Reward: TM Nature Power
Route 8 - Ace Trainer Eileen
Pokemon: Level 21 Wingull, Level 22 Fearow
Reward: TM Sky Drop
Mount Hokulani - Veteran Akira
Pokemon: Level 30 Absol
Reward: TM Snarl
Route 12 - Collector Andrew
Pokemon: Level 28 Ledian, Level 29 Pinsir
Reward: TM Psych Up
Route 15 - Trial Guide Katrina
Pokemon: Level 34 Klefki, Level 34 Skarmory
Reward: PP Max
Ula'ula Meadow - Golfer Dean
Pokemon: Level 34 Hariyama, Level 34 Alakazam
Reward: Flame Orb
Poni Wilds - Dancer Julia
Pokemon: Level 43 Oricorio, Level 43 Oricorio, Level 43 Oricorio, Level 43 Oricorio
Reward: Focus Sash
Poni Plains - Ace Trainer Cole
Pokemon: Level 56 Drampa, Level 57 Goodra
Reward: TM Quash
Poni Gauntlet - Captain Mina
Pokemon: Level 61 Klefki, Level 61 Granbull, Level 61 Shiinotic, Level 61 Wigglytuff, Level 61 Ribombee
Reward: Bottle Cap