A battery is an electrical power source which is often portable.
Batteries are electrochemical devices. That is, they store energy in chemical form, and can convert it to electrical energy when required. They are available in a very wide variety of configurations, producing different voltages, with different power output capabilities.
Some are single-use devices which are disposed of when depleted. In consumer products, these are often referred to as alkaline batteries. Others types of battery may be recharged repeatedly for multiple uses, such as lithium ion or lead acid batteries.
This tag should be used for questions about using batteries with Arduino devices. This can include using batteries as a power supply, but may also cover projects where the Arduino controls the batteries in some way (e.g. as part of a recharging circuit).