Fight Challenges are for using each character's special moves, different kinds of attacks, etc. To check your progress, press start, go to Fight Archive, Fight Challenges, and scroll to Jago. It will show you progress bars for each fight challenge.
To learn how to use a character's special attacks, choose Command List from the start menu, and scroll over to Special Moves.
For example, a fight challenge for Jago is to use ten Endokukens. That's a fireball move performed by down, down-forward, forward, and a punching attack (X, Y or RB).
I got 3 fight challenges each for: 100 Laser Swords, 10 Supreme Victories, 1000 Combo Hits, and 50 Juggle Hits.
This cannot be done in practice mode. Versus mode on an easy difficulty, rematching the same opponent is probably the quickest way.