The most used avr chip for Arduino's is the ATmega328P.
This chip is able to set a clock divider while running. So it can set it's own clock to a lower rate.
But if you do so, you mess up a lot. The baudrate is no longer valid, the millis() function is no longer valid. All pwm and tone functions are no longer valid.
As silasmoeckel wrote, power saving is done with the sleep mode. You can achieve the maximum power saving with that. As far as I know, there is no Arduino library for it. So you have to read the datasheet of the ATmega328P to see what the different sleep modes do.