From the Super Mario Wiki
Stylish Moves are extra action commands in Paper Mario: The Thousand-Year Door and Super Paper Mario that accompany every attack in the game. It is up to Mario, Goombella, Koops, Flurrie, Yoshi, Vivian, Admiral Bobbery, and Ms. Mowz to use these moves by themselves. When Stylish Moves are executed, the crowd in front of the battle stage becomes even more wild, providing more Star Power than a normal Action Command to use for special moves. Some attacks have multiple stylish moves.
To do a Stylish Move, the player must press at a certain time during an attack. Buying the Timing Tutor badge from Charlieton in Rogueport reveals a ! when the player should press , and this (along with talking to Eddy the Mask) is the only in-game way to learn all the Stylish Moves.
Super Paper Mario Stylish Moves
In Super Paper Mario, Stylish Moves are completely different. All characters have the same stylish moves. The player can chain stylish moves in this game, going from "Nice" to "Good" to "Great" to "Wonderful" to "Excellent". There are various ways to do stylish moves. They all begin by jumping on an enemy. Note that because Mario has no SP in this game, Stylish moves simply serve to give the player more points, and thus level up a little faster.