Powering the Arduino Expansion Board using a Li-ion battery plugged to J2 will power only the compute module, the rest of the board like the ADC, Arduino digital pins, etc. won't be powered. If you apply power to J2 make sure you are using the correct polarity because you could damage the compute module. You should see LED activity only on LEDs DS1 and DS2 when you apply power through J2.
Thanks Diego for the information!
So if I want to power the whole thing (pins, sensors, etc.) using battery, I should use a 9v battery (connected to J1)? I want the battery last as long as possible (e.g. a month or at least a week), but I don't think 9v battery will last that long when Wi-Fi is turned on.
Or maybe I can use a battery like the following (using usb port)? Thanks again for the help! - Weishan
To power the entire Arduino Expansion Board the power should be provided through the barrel jack or the USB client port. I strongly recommend you to check this thread: What would it take to power the Arduino Breakout from a 3.7v lipoly battery?. It is about the same question you have. There are some interesting pictures and suggestions that you will find very useful.
That's exactly what I was looking for, thanks Diego for the link!
Looks like I can still use my 3.7v battery, just need to buy a 3.7v to 5v converter like the following,