Cigstar-Mini-EGO-Twist-Battery-Variable-Valtage-E-Cig-3-2-4-8V Ecig Batteries are  power units that can contain a rechargeable lithium ion battery and this makes up the largest component of an electronic cigarette.  Although some other battery known as "mods"  use a insert-able battery unit, and mods usually have a larger capacity (explained below). Most models use a power switch button which a minority will have an electronic airflow sensor and actively triggered  by drawing breath through the device.  Usually models will have an LED light either on the button, screen (if it has one) or at the bottom to let you know it is being activated.
eVic A battery's  capacity is determined in mAh (milliAmp hours), typical ego style batteries will range from 650 mAh  up to 1300 mAh.  Smaller types can be as low as 100 mAh and larger mod and insertable batteries like the eVic, Vamo and many others may go up to 1800 mAh.  How long they will last before a charge is needed depends on  several factors such as the battery capacity, how active you are vaping and the voltage which you are using to activate the atomizer.  Some models offer a variable voltage switch which normally will range from 3.0 volts to 5 volts.  The higher the voltage, the more intense the throat hit or "vapor" hit will be, but consumption of both the battery charge and the E-liquid becomes higher. 
e-cig usb cableCharging the battery can be done with a number of methods depending on the battery type.  The ego style batteries use a usb wire that screws on to the top battery contact and then plugs in to a wall charger.  Others devices which also use the USB like some Joytech models might use the pass-through method which inserts into the bottom of battery like a phone connection. Larger mod batteries for the eVic, Vamo, or mechanical mods will require taking out the battery and plugging it into a wall charger.