For this achievement, you need to defeat all 4 Omens that Gargos will send when corruption in one region reaches 50%.
As you progress in your playthrough, corruption will gradually increase in the different regions whenever you don't complete a fight in that region (be it because you lost or you just skipped it). Occasionally, after the end of a mission you'll have to choose to do one of a set of actions and as a result corruption may increase (or decrease) in any given region.
When corruption gets to 50% in one region, Gargos will send an Omen that you will have to fight after one more turn (so, if corruption gets to 50% in, say, Asia in turn 15, you'll have to fight Omen in round 17). Whenever an Omen appears, you can send your 3 fighters if you wish (but you don't have to if you don't want to).
Each Omen carries one of the buffs Gargos will be using when you fight him, so if you defeat that Omen, Gargos won't be able to use that particular buff (and you want that). You can remove a total of 4 buffs, which is the requirement for this achievement, which means you will have to beat 4 Omens.
For me, the best way to approach this is to let corruption increase in one region up to 50% while controlling it in all others. To do this, you'll have to choose which missions to complete and which ones to skip by using "End Turn" when you've faught all the fights you wanted to fight. Once you get to 50% corruption in one region and get the warning that Gargos will be sending an Omen in one turn, you can start finishing missions in that region to decrease corruption there and start building it up in another region (even before fighting the Omen). This "controlled corruption" method allows you to have your fighters in good shape without using lots of resources (thus saving them for Godlike difficulty).
You can of course skip everything and fight 3 Omens in one turn if you want. Just take into account that if you get all 4 Omens in the same turn, you won't be able to fight one of them and therefore you won't be able to unlock the achievement. Also, be careful not to let any region go straight to 100% corruption. For example, if you are almost at 50% in a region and skip one Emergency Portal and a mission in that region, you could potentially go straight to 100% corruption, lose the region and not get the chance to fight Omen.
Finally, once you beat the fourth Omen, don't panic if the achievemnt doesn't pop. You'll need to complete another mission for it to register (even if the turn ends and a new one starts).