There are 4 achievements in the game tied to elemental kills, now this can be a combination of kills with weapons, grenades and barrels that are scattered around the maps.
You can tell if your weapons and grenade modes have specific elemental effects as they have an icon on them:
• Red Flame - Incendiary
• Blue Lightning Bolt - Shock
• Yellow Explosion - Explosive
• Green Radioactive Symbol - Corrosive
The barrels are also colour coded:
• Red/Orange - Incendiary
• Blue - Shock
• Yellow - Explosive
• Green - Corrosive
The easiest enemies to kill will be unshielded enemies like Skags, Rakk, Spiderants and lower level bandits, however it doesn't matter if they have a shield if your weapon is strong enough.
Get 25 kills with a combination of weapons, grenades and barrels and the achievement is yours.