Why is water really wet?

I am guessing that by wet, you are referring to several things: coolness, dampness, lubrication, and the ability to saturate things or make things wet.
The coolness of water is caused by a few things. First, water is a better conductor of heat than air. If you put a thermometer into a cup of fifty degree air and fifty degree water, the one in the fifty degree water will drop faster. Second, water can evaporate from the skin, absorbing energy from the skin to change into a gas. This is how sweat works. Third, water can get more surface area on your skin than most solids. Water can get into all the pores and folds of skin, while solids are left with much less on the surface.
Dampness is when water changes the properties of other things by being in them. For example, when a paper towel is damp, it gets floppy and limp. Damp hair clings together better. Damp skin is more flexible. This is caused by dehydration of dried out fibers, allowing greater flexibility.
Lubrication is from the ease in which water molecules slide past each other. By having at least two layers of water molecules between sliding surfaces, they glide effortlessly.