The Hall of Fame is where you can view the Pokémon used while facing and defeating the Elite Four and the Champion. Its data can be messed up after encountering 'M or MissingNo, but only in Red and Blue.
You can get into the Hall of Fame in just about all of the Pokémon Games. Whoever is the Champion you have beaten will take you in there to record your data of the Pokémon, their nicknames and levels that you used to defeat the Elite Four.
The player's own Hall Of Fame records can be accessed via the PC. In later games the date of their first Hall Of Fame entry is featured on their Trainer Card.