Monsters in the Battle Arena can be either Bosses or Non-Bosses. Boss monsters will always appear with purple text, whereas Non-Boss monsters will have either yellow (weak), white (normal), or red (strong) text.
Most zones have several Non-Boss monsters along with 1-3 Bosses, which are the strongest enemies in the zone. When deciding on where to Idle or Anti-Idle, the boss monster will often be the deciding factor. For example, in the Advanced Training Zone, the boss is Master Dummy. The Master Dummy has approximately 10x as much HP as the other monsters in the zone, as well as higher attack and defense and other stats. Even though you may be able to idle the rest of the monsters in a zone, the boss of the zone may kill you if you are not careful.
On the stat page, your damage may be different for Boss Damage and Non-Boss Damage, depending on skills, Orange (B) bonuses, and Invisible Ally bonuses. Because the most difficult monsters in the game are all Bosses, Boss Damage is a very desirable Orange Bonus. Note that the 'Damage' orange Bonus will give only half the bonus of the 'Boss Damage' bonus, but to both Boss and Non-Boss Damage. Because the boss is always the hardest enemy in a zone, the Boss Damage bonus is nearly always preferred to Damage, since the non-bosses are already easier to kill. In some specific cases, Non-Boss Damage is also a good bonus, such as in the Dark Pyramid or Spooky Crypt.
Bosses have better drop rates and higher rewards than Non-Bosses.