For each schtick spent in this ability, you can select one specific gun as a signature weapon. Your character might have his lucky Glock, the combat shotgun his grandmother gave to him as a coming of age present, his collector's edition ankle holster .32, the Hammer of God given by superiors in the inquisition to fight demons, and so on.
A signature gun has a damage rating of your Agility +5 if this would be better than the weapon's normal damage rating. The weapon is a part of you to such a degree that it is your aim, not the caliber, that determines the damage you do.
This schtick comes in two versions - the unique weapon, and as weapon specialization. If you pick the unique weapon, the schtick applies to a single, actual weapon, not to all identical weapons; your lucky Glock gives you the bonus, but any other Glock of the same model doesn't do anything special for you. Unique Signature Weapons have a form of script immunity, a villain may steal if from you, but it will always pop up somewhere later in the story. It will never become permanently lost or destroyed unless the GM develops a plot around it. They never break down, though they can still jam. A unique signature weapon is very precious, and can be held hostage if someone manages to steal it.
Simple Weapon Specialization means you get the bonus with any weapon of the specific model. It does not give the weapon script immunity, but if you do lose your gun, you can replace it with an identical model.
It is harder to take any signature weapon away from you - you get a +3 bonus to resist any stunt aimed at harming or taking away your weapon. Customs officials and doormen often turn a blind eye to them. Unless the weapon couldn't be concealed under the clothes you wear or the security inspection merits a scene of its own, you always manage to smuggle a Signature Weapon on board airplanes, into presidential palaces and so on.