This was actual the very first thing we learned in my Biology class last year. Water is used for mainly three things: coolant, medium for metabolic reactions and transport medium.
Why is water a coolant? Well water has a high specific heat. This means that water can absorb or lose a relatively high amount of heat without changing temperature much. Water also has a high heat of vaporization. This means when water evaporates, it brings a lot of heat with it (and that heat is last to the original organism). These should show you that water has a pretty big role in stabilizing body temperature, and also cooling. Because of its high specific heat, and because most organisms are made up of a large percentage of water, organism's core temperature will not fluctuate greatly (which, if it did, could cause enzymes to denature and a bunch of other bad stuff). Also, because water takes so much heat with it when it evaporates, it acts as a coolant. Think of this as perspiration. When humans perspire, they are getting rid of heat to maintain a stable, relatively normal body temperature.
Next, water's polarity and cohesive property are very important. Cohesion implies that water molecules will tend to 'stick together'. This concept is very important to the way in which water is able to defy gravity and push through roots to the very top leaves in trees (and other plants). Cohesion also allows water surface tension, such as on a pond, and thus allows certain insects to 'walk on water' without sinking.
Finally water is an excellent solvent for other polar molecules. Many other molecules are also polar, especially those related to organisms: carbs, proteins, nucleic acids, etc. These molecules are travel very well in water (polar molecules attract each other essentially). However, note that lipids are usually non-polar and that's why oil and water don't mix. Hence, water acts a great transport medium for most molecules of organisms. In addition, it provides an aqueous environment for which most biochemical reactions can take place (metabolic).