A pokémon professor is considered an expert on Pokémon in his or her respective region. Pokémon Professors are often researchers as well, and many times they are the ones responsible for distributing Pokédexes and giving the player their first Pokémon.

According to a comment Professor Birch makes when grading a player's Pokédex, catching a large number of Pokémon species helps to qualify one as a Pokémon Professor. However, as Professor Oak mentions he has not completed the Pokédex, this is presumably not the only way to become one.

The pokémon professors and their areas of study include:
Professor Oak of the Kanto region, who studies the relationships between Pokémon and humans
Professor Ivy of the Orange Islands, who studies the regional differences between Pokémon
Professor Elm of the Johto region, who studies Pokémon breeding patterns
Professor Birch of the Hoenn region, who studies Pokémon habitats
Professor Rowan of the Sinnoh region, who studies Pokémon evolution
Professor Krane and Dr. Kaminko of the Orre region, who study the purification of Shadow Pokémon and the creation of Robo-Pokémon, respectivey
Professor Hastings of the Fiore and Almia regions, who created the Capture Styler, used by Pokémon Rangers

Professors from regions in the main series are all named after trees.

-Defiinition from Bulbapedia

No competition here, Belle would easily make the best pokémon professor. I could see her researching in a lab or in the field and giving trainers their starter pokémon. Tell me what starter pokémon she gives out and I'm going to her region!

or Pocahontas!