Dexterity measures agility, reflexes, and balance.
Some of the things which a character's Dexterity score affects are:
Armor Class- When a character is capable of reacting to an attack, they add their Dexterity bonus to their Armor Class (AC), making them harder to hit. Note that some types of armor restrict how much your Dexterity score can improve your Armor Class.
Ranged Attack Rolls- Characters add their Dexterity modifier to their attack rolls with ranged weapons (i.e., bows, crossbows, darts, slings, etc.) and Ranged Touch Attacks. Additionally, Swashbucklers and other characters who take the Weapon Finesse feat can add their Dexterity Bonus to their attack rolls with light weapons (rapiers, short swords, etc.). Note that a character's Dexterity bonus does not typically modify the damage rolls for these weapons.
Reflex Saving Throws- Characters add their Dexterity bonus to their reflex saving throws, used for things like avoiding fireballs and other things which you can escape by moving quickly.
This ability is most important for Rogues, Warlocks, and Swashbucklers, but also for characters who typically wear light or medium armor (Barbarians and Rangers) or none at all (Monks, Wizards, and Sorcerers), or for any character who wants to be a skilled archer.