We have all seen what happens after vinegar and oil salad dressing are vigorously shook; one does get a mixture of sorts, but after a little time the ingredients separate with the lighter oil on top and a denser vinegar/water solution on bottom. This is an illustration of an important chemistry principle expressed by the rule that 'like dissolves like.' This refers to the phenomena that when two liquids made of molecules of similar size and polarities are mixed, they will usually form a single phase solution, no matter what the relative number of moles of each species.
This is expressed by the jargon that the two substances are miscible in all proportions. In contrast, when a highly polar substance, such as water, is mixed with a nonpolar or weakly polar substance, such as most oils, the substances will separate into two phases. This phenomenon is usually rationalized in introductory chemistry text books by saying that oil is hydrophobic, and thus does not make solutions with water, while polar small organic acids (such as acetic acid from which house vinegar is made) are hydrophilic, and thus are miscible with water.
This explanation almost universally leads students (and even some professional chemists) to believe that individual water and oil molecules repel each other, or at least attract each other very weakly. Nothing can be further from the case! An individual oil molecule is attracted to a water molecule by a force that is much greater than the attraction of two oil molecules to each other. We can observe the consequence of this greater attraction when we put a drop of oil on a clean surface of water. Before hitting the surface, the oil will be in the shape of a spherical droplet. This is because the oil molecules are attracted to one another and a spherical shape minimizes the number of oil molecules that are not surrounded by other molecules.
When the oil hits the surface of the water, it spreads out to form a thin layer. This happens because the attractions between the oil and water molecules gained by spreading over the surface is larger than the oil-oil attraction lost in making a large oil surface on top of the water. If a sufficiently small drop of oil is put on the surface, it will spread to form a single molecular layer of oil. By measuring the area produced, one can get a simple estimate for the size of each oil molecule and thus Avogadro's number. Given these strong interactions, why does not each oil molecule dive into the water solution and surround itself with the favorable water attractions?
The reason is that to do so, it must come between water molecules that are already attracting each other! The strength of water-water attraction is much higher than water-oil interactions, and thus there is a net cost of energy in putting the oil molecules into a water solution. Thus the vast majority of oil molecules stay out of the water, though as many as will fit will hang on to the surface water molecules that do not have a full complement of partners.
In the end water and oil will not mix to form a solution because a non-polar chemical like oil will not dissolve in water. However, with the use of a surfactant, oil and water can be made to form a stabilized suspension that is uniform enough to be very similar to a solution. Surfactants have many applications that involve mixtures of water and non-polar chemicals like oils.