Of course starting with cold water...
The most Fuel efficient way is the stove. Look at this way, electricity is generated by heating water and running the steam through a generator. . When you do that you loose energy in the form of heat, transportation and so on. They use coal, or natural gas to power such a plant. The Stove usually uses natural gas or something similar, but in essence the same primary energy source as needed to make electricity. Therefore, when using electricity to heat water you are repeating a process where energy loss occurs at least twice. The stove doesn't need transformed energy. And is therefore (much) more effiient.
An electric boiler will be more efficient than a pot you put on a stove, because much of the heat from a stove burner escapes around the sides of the pot, especially with a natural gas or propane stove. So using an electric boiler is better than using a pot on an electric stove, and is probably more energy efficient than boiling a pot of water on a gas stove as well. In the case of a gas stove, an electric boiler could be less efficient in terms of both price and greenhouse gases produced, if your electricity comes from coal or natural gas, because only about a third of the heat from burning the fossil fuel at the power plant reaches your house as electricity, whereas over 60% of the heat in the natural gas or propane actually heats up the pot of water on your stove. Of course, if you switch to a green electricity supplier, you won't produce any greenhouse gases at all by using your electric boiler. You can probably have a bigger impact on the cost of boiling your water, though, if you're careful to heat only the water you need, only for as long as you need it, and only to the temperature you need it. See the article below for further tips.
