In the context of residential solar+storage systems, a hybrid inverter (sometimes referred to as a multi-mode inverter) is an inverter which can simultaneously manage inputs from both solar panels and a battery bank, charging batteries with either solar panels or the electricity grid (depending on which is more economical or preferred). Their capabilities may go beyond this however – some devices also handle inputs from wind turbines, generators and other power sources.

Pros:

All-in-one inverter solution for grid-connected solar-plus-storage systems
Frequently intelligent and programmable for maximising overall system efficiency and savings
Can usually be installed without batteries for future expansion
Long history of use in off-grid and stand-alone power systems
Cons:

Less design flexibility than modular solutions which use separate PV and battery inverters
Generally less efficient than dedicated solar-only or battery-only inverters