Electronic cigarettes contain a few different different components.
The battery is what sends power to the heating element when you press a button. This power will heat up the e-liquid to the point that it turns to vapor, which is then inhaled by the user. These devices can often be referred to as "mods." Each mod will contain a few features: buttons to change power settings, a power button to turn the device on or off as well as to fire the device, and a 510 connection which the "tank" or "atomizer" will screw into. The battery in your device can either be internal (meaning that it is built into the device and not replaceable) or external (meaning that you can take out the battery and put a fresh one in.)
The atomizer (sometimes referred to as a tank) is the part of the device that contains both the coil and the e-liquid. Some tanks have a reservoir that holds enough e-liquid so that the user doesn't have to refill it constantly. Some atomizers called "drippers" are devices that contain an exposed coil that the user will drip e-liquid directly on to. These types of devices don't have reservoirs.
The coil is the heating element of your device. These are often replaceable units that contain wire and a wicking material (usually cotton) to absorb the e-liquid that surrounds it. When the user presses the button on their battery, the coil will heat up the liquid that has been absorbed by the cotton and turn it into vapor. Some devices require that a coil be built by the user.