Eye for an Eye quests are similar to Assassinations, in that, they will require you to kill commanders, but you will have to do so with a specific weapon type, like a sniper rifle or a knife, as examples. Also, another difference is that stealth isn’t required at all, so you can just go right in and start blasting away enemies with no regard to your own health.
Note that the weapon you’re told to use to kill the commander(s) is just for them, as you can use other weapons on the regular enemies. Also, the weapon named is just the type, so if you’re to use a sniper rifle, you can use any weapon that belongs in that class, like the Z93 or the SVD. One last thing is that you will always have to go to a spot before heading to the commander(s), where you will be able to grab a weapon of that type.
There’s a total of eight Eye for an Eye quests in the game and the only rewards for doing them will be Rupees bonuses for completing three, six and all eight.