Actually, the most common usage wouldn't relate to oil and water.....
Decanting more usually relates to liquids that have a small amount of solid residue contained with them. The liquid is allowed to stand so that the solid falls to the bottom of the container, and the liquid is then slowly poured into a clean vessel so that the solid residue (and a little of the liquid) remains in the original container.
It has often been used to separate residue from wine before drinking it. But you can also use it to separate off solid precipitates from liquid solutions (although they may require some centrifugation first in order to speed the process of sedimentation).
Oil and water would not generally be separated by decanting, as tilting the container runs the risk of the lower (water) layer still finding its way to the spout along with the oil. Instead, the mixture is placed into a separating funnel, allowed to separate into two layers, and the lower (water) layer is run off by opening a tap at the bottom of the funnel.