Staggering monsters by exploiting their "weaknesses" is a mechanic in Final Fantasy XI that was introduced in the Vision of Abyssea update. The purpose of staggering a monster is twofold; first, it prevents the monster from performing certain actions for a short period of time, and second, it increases the rewards (or chance of rewards) for defeating the staggered monster.
The mechanics of staggering a monster vary depending on the battle system, and sometimes within each battle system. In Abyssea and Voidwatch, staggers are chosen from a set list of procable spells and abilities; in Dynamis, all abilities, spells, and weapon skills are capable of procing, but only on specific monsters at specific times. Boss monsters in Walk of Echoes can also occasionally be staggered, but the mechanic behind this is unknown and may essentially be completely random.
Details about how each system works can be found on the following pages: