To unlock the Facemelter trophy in Borderlands: Game of the Year Edition, you will need to kill 25 enemies with corrosive weapons.
Corrosive weapons can be identified by the green hazardous chemical symbol in your inventory (similar to the trophy art image). You can also do corrosive damage by shooting the green barrels.
When you use a corrosive weapon, green numbers will appear around an enemy to indicate that you are causing corrosive damage.