Weapons can be found, won or bought in FFVII. They usually increase a characters stats, however, less powerful weapons will decrease stats compared to newer, more powerful weapons. Each character has their own particular weapon type which only he/she can use. They also allow you to equip more Materia with their slots, (unless there are no slots) and gain more AP for leveling up your Materia (unless the AP is 'Nothing').
Weapons can have different four different AP types. Normal, Double, Triple and Nothing. These come into use after a victorious battle when you gain your EXP and AP. Note that the effects of the AP type equipped to the character's weapon only affects the character and not the whole party. Same goes for any status or elemental effects a weapon may have.

AP	Effect
Normal	Character gains normal AP amount.
Double	Character gains double the AP amount.
Triple	Character gains triple AP amount.
Nothing	Character gains no AP.