Is surface tension a property of water?
It definitely is a property that comes with water, but it is not a property that is unique to water.
Liquids are held together by intermolecular forces. These forces are not as strong as they are in solids, so they allow for the molecules to flow a bit, but they are not as weak as they are in gases, so liquids have defined volumes.
Now, molecules experience intermolecular forces between each other. This is illustrated by the diagram below:
Notice how each molecule is being pulled both outward (towards the walls), and inward (towards the rest of the molecules). The net inward pull experienced by the surface molecules is what defines surface tension.
Think of stretching a rubber band -- to counter your stretch there is an equally strong inward force exerted by the rubber band, which is why it feels so tensed. This is what is happening to those surface molecules.
It is also worth noting that surface tension increases with greater intermolecular force strength.
Water has a pretty high surface tension, because the hydrogen bonds between its molecules are pretty strong. This is what allows cool stuff like this to take place:
However, it is not a property that is unique to water.
Hope that helped :)