To get this achievement, you need to complete the entire "Chains of Promathia" expansion pack storyline, as well as all of the "Rise of the Zilart" expansion pack storyline.
Once these are both complete you can start the quest "Shadows of the Departed" which leads to the last fight of both expansion packs "Apocalypse Nigh".
Winning this fight will grant you the title of "Breaker of Chains"
Since the level cap in FFXI has been raised to 99, this achievement is soloable by certain jobs but as with everything it is easier and more enjoyable if you have a few friends to help out!
Wiki link below for more info;http://wiki.ffxiclopedia.org/wiki/Apocalypse_Nigh