There are a number of urban legends surrounding the way that water boils. It's important to keep the facts separate from the myths, though, in order to have the most accurate idea of how to best perform your cooking tasks. The idea that cold water boils faster than warm water is a fallacy; the opposite is actually true.
The process of boiling water occurs when the temperature of the water reaches its boiling point. The natural boiling point for water is 212 degrees Fahrenheit (100 degrees Celsius). As you keep your water over a heat source like a stove, the heat transfers to the water at a direct rate. This rate is defined by the principle that one calorie of heat energy raises one milliliter of water by one degree Celsius (which is about 1.8 degrees Fahrenheit). Thus, the longer that you hold the water over the heat, the more of the water will be heated. However, the rate at which it heats up doesn't change, provided that you keep the water in the same place and don't adjust the heat source.
Why Warm Water Boils Faster
Warm water boils faster than cold water because it has less heat difference to make up in order to reach the boiling point. Cold water starts out at a lower temperature and must spend a certain amount of time over the heat source in order to reach the heat that the warm water is at from the beginning. This advantage makes it easier to boil warm water than cold water, and ensures that, all other conditions being equal, warm water will boil faster than cold water.
A Related Legend
One of the other most common urban legends that relates to the myth of cold water boiling faster than warm water has to do with salt. You'll often hear of people throwing a pinch of salt into a pot of water in order to encourage it to boil faster. This legend is actually true; salt will help your water to boil faster. However, many people don't properly understand why this is the case.
Adding salt to your water doesn't change the rate at which the water heats up. Rather, it affects the boiling point of the water itself. In effect, the salt lowers the boiling point of the water, meaning that it will begin to boil before it heats to 212 degrees Fahrenheit. This means that it will take less time for the steady heating of the water to reach that boiling point, and your water will boil faster. Still, most people find that the amount of time that adding salt will change your water boiling by is relatively small.