Reducing Arduino Power Consumption
Alex the Giant 
Removing Extra Hardware
Every integrated circuit (IC) needs power to function. By reducing the number of ICs needed, you can save a bit of power. The Arduino Uno and RedBoard both have a USB bridge that converts the USB signals to signals that the Arduino's Universal Asynchronous Receiver Transmitter (UART) can use. The FT231x, used on the RedBoard, alone draws around 10mA. Just by replacing the SparkFun RedBoard with a 5V Pro Mini, not only do you save quite a bit of physical space, you can reduce the supply current from 15mA, down to just 4mA.
Ditching the Linear Regulator
Linear regulators are great. They're cheap and can regulate a higher voltage to a lower voltage with as few as 3 pins (Vin, ground Vout). The downside of a linear regulator however is they can get pretty hot when you have a large difference between the input and output voltage, or if you're drawing a lot of current. You can calculate the power that gets wasted in the form of heat with a simple equation:
With a linear regulator, the input current is equal to the output current. Linear regulators are, at best, only around 70% efficient. The more current you draw, the lower the efficiency.
Without going into too much detail, switch mode power supplies are much more efficient because the input current doesn't have to be the same as the output current. With a light load, you can find switching regulators that are over 90% efficient. With a switching regulator, you can not only reduce the input voltage (using a buck converter), but also increase the voltage (using a boost converter). Other ICs, like the TPS61200 used on the PowerCell, are perfect for battery powered applications, because it can efficiently reduce the battery's voltage or boost it!