Instead of following the relatively unsophisticated brute-strength approach employed by warriors, characters who can Sneak Attack - typically Rogues - instead employ finesse and ruthless efficiency to punish vulnerable enemies. Against a foe whose defense is compromised they can add bonus Sneak Attack damage, a benefit that represents the attacker has struck a vital spot.
Attackers can apply Sneak Attack damage any time the target is denied their Dexterity Modifier to Armor Class... regardless of whether they have a Dexterity Modifier worth applying is irrelevant. This most commonly occurs when the defender is Flat-Footed, but characters who are held, paralyzed, sleeping or caught in a Web spell also are vulnerable to Sneak Attacks. An attacker can also deal their Sneak Attack damage to defender they're flanking, and this is quite possibly the most common application of this form of damage. A melee-inclined Rogue would be well-served by investing in Mobility so they can sneak into flanking positions in relative safety.
Sneak Attack damage is typically a variable number of d6 rolls added to whatever damage the attacker normally deals, and it can be added to any attack - even to multiple attacks per round! Suffice to say, while a Rogue may normally not be a very impressive combatant, adding several d6's of damage to their attacks changes things quite a bit.
There are some limitations, however. This damage is not multiplied during critical hits, and a Rogue must be within thirty feet to add Sneak Attack damage to a ranged weapon (including spells with Range Touch Attack rolls - yes, even something as humble as Acid Splash can be used to Sneak Attack). An attacker can also not apply Sneak Attack damage to an opponent which benefits from any form of concealment, whether it's from being shielded by terrain or from the effects of spells like Blur, Displacement or Invisibility.