A hybrid inverter (or "smart inverter") functions as both a battery-based and a grid-tie inverter. In a solar power system, a hybrid inverter provides the benefits of grid interconnection (like net metering to keep those bills from the power company low) without its drawbacks (like losing power when the grid goes down). A hybrid solar inverter is able to do this by intelligently sensing when the grid goes down and then powering critical loads in the house with a connected battery bank.
Some hybrid inverters - like the Schneider Conext XW models - even have a second AC input, which is commonly hooked up to a generator that can keep your battery bank charged while the grid is out. A hybrid inverter manages all these transitions automatically so that your critical loads never lose power - no wonder a hybrid inverter is also known as a "smart inverter"!