From Bulbapedia, the community-driven Pokémon encyclopedia.
Pokédex holder is the term used by Professor Oak in the Pokémon Adventures manga to describe a Trainer that has
been given a Pokédex. These Trainers are often very gifted in one particular area, which Professor Oak has dubbed each’s “special skill”. These Trainers all have “starter” Pokémon from their regional Pokémon Professors (although the Pikachu Oak gave to Yellow was Red’s and she later caught her own), though most already owned or trained Pokémon prior to receiving them. The Pokédex holders have become a very tight knit group with many of them bonding with their cohorts in their regional adventures and later with their seniors and juniors.
In many cases they have been targeted, due to the fact that they were powerful (especially as a group) or simply that they had a Pokédex in the first place, making them quite vulnerable. During the FireRed & LeafGreen chapter, the Kanto Pokédex holders, plus Silver, were turned to stone. Months later, they were revived by Emerald.
The Pokédex holders are named for 15 main series games that are not remakes, with the exception of Cheren and Bianca. Also, White is not a pokedex holder despite being named for a main series game.