All elemental weapons can do splash damage that affects any targets in that area
Additionally damage over time can spread from one enemy to another nearby, e.g. if a bandit gets close to another bandit who's on fire, there's a chance that he'll catch on fire too.
Explosive damage behaves slightly differently as it has an instant area of effect. I don't think other elements on area of effect weapons such as grenades and launchers cause additional splash damage on top of the AoE, but the DoT effects certainly can spread.
Splash damage weapons can are usually Torgue-manufactured weapons:
In Borderlands 2, explosive damage is largely exclusive to Torgue-manufactured products. However, several other manufacturers feature weapons that deal explosive damage in some manner. They include:
- Assault Rifles that use Torgue barrels (Bandit, Dahl, Jakobs, and Vladof)
- Rocket Launchers (Bandit, Tediore, and Vladof)
- Cobra sniper rifle
- Tediore non-elemental weapons (when reload-thrown)
- Bouncing Bonny, Bouncing Betty, Jumpin Bitty, and Jumpin Biddy grenades
- Grenades and Gurnades
- Murrv, Mrvur, and Muvr grenades
- Kiss of Death grenade
- Singularity Grenades
- Transfusion Grenades
- The Cradle shield (when depleted and discarded)