The Master is a martial arts master who resides in the Dojo of Toad Town.
In Paper Mario, after beating Chan and Lee, Mario can fight the Master. If Mario loses, the game does not end, as it is not a fight to the death. After being defeated once, the Master can be fought again, and powers up. He can also be fought a third time, and is even stronger than the final form of Bowser. Once defeated, the Master gave Mario the Diploma, to indicate that Mario is now the strongest. The Master does not fight Mario again after this.
The first time, the Master has 50 HP, 6 ATK, and 0 DEF. His only move is to punch. At the point near the end of the game he is fairly easy, easier even than Huff N. Puff and the Crystal King, but if fought too early in the game he is hard, maybe even unbeatable.
The second time, the Master has 75 HP, 8 ATK, and 0 DEF. Aside from his punch, he can also do a combo, charging at Mario for 5 damage and then uppercutting him for 6. This can be countered with the Zap Tap Badge, which can be bought from Merlow at Shooting Star Summit, or by eating a Volt Shroom (either electrifies Mario so enemies that make direct contact get zapped, and any combo attacks are interrupted).
The third time, the Master has 99 HP (apparently the maximum enemy HP in Paper Mario, as Bowser has this amount of HP and heals for 30 HP three times to make it seem like he has more HP). He has an ATK of 10 and DEF of 1. He retains his punch attack and his dashing uppercut (for 6 then 7 damage this time), and does an even tougher combo where he hits Mario for 5 damage, then 6, then 7, for a total of 18 damage. The Zap Tap or Volt Shrooms can stop this attack so that Mario only receives 5 damage from it.