Simply put, easterly trade winds over the equatorial Pacific Ocean are partly to blame for both phenomenon. For La Nina, the easterly trade winds strengthen. This blows more warm water west, and allows cold water below the ocean's surface to push towards the top near the South American coast to replace the warm water.
In an El Nino, the opposite occurs. The easterly trade winds become weaker, and can even reverse direction. The warm Pacific Ocean becomes nearly stationary or pushes eastward and gains heat. Besides affecting weather, El Nino has also been known to hurt fishing off the coast of Peru.