The Ninja perk grants you the power of the fabled shadow warriors. When attacking you gain a +15% critical chance on every strike, only with either Melee or Unarmed. Sneak attack criticals do 25% more damage than normal. This perk is more suitable for players who follow a "ninja" play style, similarly to Grim Reaper's Sprint, which is suitable for players who follow play styles that use V.A.T.S. more often.
Notes
To attain this perk you are only required to have an 80 Melee Weapons skill, even though the perk affects the critical bonus and sneak damage of both Unarmed and Melee Weapons.
The extra 25% sneak attack critical damage and 15% critical chance only applies to melee and unarmed attacks. Attacks with any other weapon type will not be affected.
Bugs
In Fallout: New Vegas Ninja multiplies your critical chance by 1.15 instead of adding a flat 15%, unlike all other perks/traits that modify critical chance. For example, a character with a critical hit chance of even 25% would have it be increased to only 28.5% rather than to 40% as it should. (25% * 1.15 = 28.75% vs 25% + 15% = 40%)
In Fallout 3 this perk is broken, or at least doesn't work as it should, for both the sneak attack critical and melee critical bonus.