The Mew Glitch (also known as Long-range Trainer Glitch) is a glitch found in all of the Generation I core series games. Due to using the same engine, it is also present in Pokémon Brown.
Explanation and Methods
Please refer to the Bulbapedia article for detailed info.
Mew Glitch in Pokémon Brown
In Pokémon Brown, most long-range trainers are embedded in walls are therefore not usable for the glitch under normal conditions. The Jr. Trainer♀ in Silk Tunnel and the Cue Ball on Route 49 are required to be fought to progress, making them not ideal for usage. Therefore, the most practical long-range trainer to use is the Youngster on Route 62.
Due to the glitch producing a trainer for values from 200 to 255 and the game having Pokémon with values within this range, it is not possible to obtain every Pokémon using this method. Notably rendering either Ho-oh or Lugia unavailable without the use of other methods or cheats.