Once you acquire the Gun Mage dress sphere, you character will earn a new type of ability which is a lot like Kimahri's Lancet skill from Final Fantasy 10. However, instead of your character being blue, carrying around a spear and doing some fancy moves to learn the ability, Final Fantasy X-2's system just requires a lady and a gun.
The Blue Bullet system is easy enough to understand. In battle, if one of your team has the Gun Mage dress sphere equipped and an enemy (only an enemy though. A team member can't teach it to a fellow team member) uses one of a series of skills on you, that character will learn the "Blue Bullet". Most of them are easy to acquire (because they're offensive skills), but some require a little more skill...