Surface tension can be briefly described as the force that exists at the interface between two media. It exists because molecules lying at the boundary between two media have more energy. An analog phenomenon is observed with penguins in freezing Antartic winters – those at the border of their group need to spend more energy in order to keep warm. The same idea applies to molecules, which “feel more comfortable” when they are completely surrounded by other similar molecules. Nature tend to minimize the number of those molecules exposed to the outside, therefore reducing the surface area of interfaces. When a boundary between two media is stretched, surface tension brings it down to the smallest surface area possible.
While this phenomenon is very fast and occurs at a microscopic level, it has many macroscopic consequences. When surface tension dominates other forces, water tends to form spheres in the air (drops) because the sphere is the smallest possible surface. Wetting is driven by liquid surface tension, but it also depends on the solid medium that the liquid attempts to wet. If a drop of water is placed on a material whose surface tension is weak (a plastic film for example), the internal forces of attraction of the water drop will be stronger; the drop will therefore be practically spherical and will minimize its contact area with the underlying solid. In this case the solid is not wet: this is what happens when rain falls on a lotus leaf and forms water pearls that bounce off its surface. On the contrary, if we put a drop of water on clean glass, the drop strongly spreads out and wets the surface. It is this same phenomenon of surface tension that explains why some insects can walk on water (even though their density is greater than that of the water): equipped with hydrofuge hairs under the legs, they never break the surface tension of the water and fall into it.
One last example: it is also surface tension that explains why… greasy plates need to be cleaned with hot water and “washing-up liquid”. It is because soap is a tensioactive carbonated chain that contains a hydrophobic side and a hydrophilic side. The latter will link onto molecules of water and decrease their “frustration” to be in contact with oil molecules. In other words it reduces the interfacial tension between oil and water, and it then allows water to disperse particles of fat more easily.