Every time you win 10 rounds in the Super mode, you will encounter a boss character. Most of the time it will be a fairly generic trainer, but every once in a while you will face off against a character from a previous game. The first time you clear Normal mode, you will be guaranteed to face off against Red or Blue, and then you will be gifted the Mega Stones for the Kanto Starters.
After you defeat them, you will get loads of BP plus you can scout them as partners to select from later if you ever intend on doing Multi Battles!
The following tables list all of the major characters you can battle.