Water and other water-based molecules are what is called "polar", meaning that the electrons in the molecule forming the molecule's bonds spend more time closer to one end of the molecule than the other. This causes one end of the molecule to become negatively charged, while the other end becomes positively charged. Because opposite charges attract, water molecules tend to line up like magnets due to the interactions between them. Most molecules that dissolve in water, like sugar, salt, etc. also are polar (which is why they dissolve so easily - water molecules can pull them apart).

Oil doesn't do this; the distribution of charge over an oil molecule is even, with no polarity. As a consequence, if you mix oil and water, the water molecules will find each-other and form these chains, which will cause them to form together into their own pockets of water, thus separating them from the oil.