How do I know if a development board can be charged by batteries?
Any board can be run with batteries if you match the voltage and current needed by the board.

For boards that can recharge batteries, you would look for boards with built in lithium or other battery type charger circuits. There are some arduino type boards with this built in,  but there are also modules you can buy to connect. 

As a kludge, you can just use a usb battery pack. Built in battery and charging and voltage regulation.