1. The oil can be burned. This works well even if the oil is floating on water, but isn't very good for any life under the oil. However, relatively small amounts may be burned to prevent spills, as happened with the freighter which recently ran aground off the west coast of the United States; the fuel oil on board was burned to keep it from spilling.
2. The oil can be skimmed. Again, oil floats on water, so it can be skimmed off in the same way that grease can be skimmed from a stew. There are several methods available for this, all of them similar to the skimmers used in kitchens. The oil is confined during skimming by a floating "boom," which goes down far enough to keep the oil from seeping out from underneath it.
3. The oil can be broken up or washed (converted into an emulsion) by the use of detergents. This is exactly similar to washing clothes or dishes. Fats and oils which don't dissolve in water are treated with a surfactant (soap or detergent) which carries the oil into water "solution."

Bio-engineered oil-eating bacteria have been suggested, but there's a problem with them: what if they find their way into oil wells? There are safeguards one can use (such as making the bacteria dependent on a nutrient we supply), but this is still probably the worst solution because bacteria are known to exchange genetic material with other bacteria of different species.