At its core, a home battery consists of battery cells (today mostly lithium-ion, often the safe lithium iron phosphate/LFP type), a battery management system for monitoring, and an inverter that converts the battery's direct current into usable alternating current. The system is typically integrated between the PV array and the house distribution board and can usually be monitored via an app.
In everyday use it works like this: during the day the solar system first covers household consumption, then charges the battery; once it is full, the surplus goes to the grid. In the evening and at night the storage covers demand before any power is drawn from the grid. This significantly increases self-consumption, making you less dependent on grid prices and feed-in tariffs.
A battery makes most sense for households with their own PV system, high evening and night-time consumption (for example with a heat pump or an electric car), or a desire for greater self-sufficiency. Without a solar system it usually only pays off in special cases.
Pay attention to the right size (an oversized battery is expensive and rarely fully used), to warranty and cycle life, to professional installation by licensed electricians, and to any backup-power function. Common mistakes are oversizing and poor coordination with the inverter and PV system.