There are two "impossible" situations in the game, both of which require you to have very high levels in charm or intimidate. Only focus on one, as putting points in the other is a waste. If you do not have enough, you will not have enough to resolve the situation. Both these decisions are made in the second half of the game, and contain spoilers for those have not made it that far.
Your first impossible situation is on Virmire, and it involves Wrex. When you find out that the facility Saren is using on the planet is to cure the genophage, and plan to destroy it, Wrex becomes extremely upset. You must talk with him, and try to reason with him as to why the facility must be destroyed. If you have 8 points in either of the alignment skills, you will be able to talk him down from his stance, and he will agree to keep on fighting, which unlocks the achievement.
The second impossible situation takes place during the finale of the game, on the Citadel. When you finally are able to get to the top of the tower, in the Council Chambers, you will find Saren. After talking with him, you must convince him that he's become indoctrinated, and that there's only one way to end it. Your skills need to be very high to unlock these dialogue options. If you are able to convince him to commit suicide, you will unlock the achievement.