What are (or how do I use) the power saving options of the Arduino to extend battery life?
Note: I wanted to point out that, even though my answer was accepted, the reader (you) should really read Anindo's answer first if you really want to save energy with any Arduino board. Once you address those items in your design, then you should read my answer and worry about how to set the MCU into low power mode.

Having said that, there are several techniques to make an Arduino save power. I'll list a few and then point you to a page that explains them all in more detail.


While the controller isn't doing anything important (between one read of a sensor and the next, for example), you can put the controller into one of the sleep modes below, with the command set_sleep_mode (SLEEP_MODE_PWR_DOWN). Next to each mode is the approximate power consumption of each mode. 


SLEEP_MODE_IDLE: 15 mA
SLEEP_MODE_ADC: 6.5 mA
SLEEP_MODE_PWR_SAVE: 1.62 mA
SLEEP_MODE_EXT_STANDBY: 1.62 mA
SLEEP_MODE_STANDBY : 0.84 mA
SLEEP_MODE_PWR_DOWN : 0.36 mA

Disable brown-out detection (the circuitry that turns off the controller when low voltage is detected).
Turn off ADC (analog to digita conversion)
Use the internal clock


Then, when you put the controller to sleep, you need to use one or more mechanisms below to wake up the controller and do something with it:


Wake up with a signal
Wake up with a timer


This is a summary I made from -


Nick Gammon's article: Power saving techniques for microprocessors.


That article applies mostly to ATmega328P, but the technique applies to other Arduino compatible controllers as well. As TheDoctor said well, you will need to check the datashet to make sure your controller suports any of those techniques and how to do it more precisely.