Here are my suggestions:Use a resistive voltage divider to reduce the (up to) about 15 V down to about 5V. 220K and 100K will form a 3.2 to 1 voltage divider, without wasting much power.Use a step-down switching regulator for maximum efficiency and bypass the barrel jack, as the onboard regulator wastes power as heat. To do that, use a 5V switcher and power the Arduino via the 5V terminal. Power the relay(s) with a separate regulator, or use the battery voltage directly for the relay power, if appropriate. Observe the maximum current ratings of the regulators.Here is a good selection of inexpensive, efficient step down regulators, from a trustworthy (non eBay) source: http://www.pololu.com/category/131/step-down-voltage-regulators
Good advice jremington. For the relays, I prefer to use relay modules that already have the transistors and flyback diodes. They are usually 5VDC so the coils can be run off the same supply as the Arduino (unless you want to energize a bunch of them at a time which may stress the power supply).One more thing -- I don't recommend using a car battery. Car batteries are designed to deliver a lot (100s of amps) of power for a short (seconds) time. For a more moderate load for a longer period -- like, for example, water pumps -- a deep cycle marine battery would be far more suitable.