It has been theorized that when one becomes a Pokémon champion of a regional Pokémon League, one attains the title of master - however, this cannot be correct, as Ash won the Orange League without attaining the position. Indeed, other League champions haven't ever been referred to as Pokémon masters, either. Another theory is that, in the context of the anime, Pokémon masters are the members of the Elite Four. However, even these characters are not always referred to as Pokémon masters when Ash meets them.
Another popular theory is that one must catch one of every kind of Pokémon in the world to become a Pokémon master. However, this is likely to be incorrect as far as the Pokémon anime is concerned as well. For one, it would be impractical, as capturing legendary Pokémon is both extremely difficult, and is often shown to be morally wrong. In addition, various negative and dangerous effects appear to come as a result of capturing legendary Pokémon. While some legendary Pokémon have willingly partnered with humans (for example, an Articuno was befriended by Noland of the Battle Frontier and battled for him on occasion), it is highly implausible that anyone has ever convinced all of them to do so.
Aside from the legendary issue, it would be impractical for anyone in the anime to capture and maintain at least 493 different creatures. Unlike the Pokémon games or Pokémon Special manga, in which Pokémon can be stored in suspended animation as computer data - Pokémon in the anime need to be fed and exercised regularly to maintain their health. Such a task would be impossible for anyone to perform alone, or even with many assistants or helpers. A worldwide community containing at least one of each is logically possible, considering that Professor Oak had a research facility which doubled as a living ground for the Pokémon. This may have resulted in the franchise's dropping of the familiar "Gotta Catch 'em All!" motto.
The answer may lie by looking at a real-life skill that also has trouble defining a master - martial arts. In that field, as in the world of Pokémon training, even winning tournaments, belts, and competitions doesn't define if someone has achieved the status of a "master". Instead, masters are defined more by inner character, inner strength, and intuition, rather than who has the most power or skills.
Indeed, a Pokémon master may not be someone who has won a title, so much as it may simply be someone who has striven so diligently, that Pokémon training and befriending Pokémon is intuitive.
According to the Electric Tale of Pikachu manga, a Pokémon master is simply an elite trainer who is considered a professional and regularly takes part in League competitions. - Bulbapedia
