Solar only systems cannot power your home during a blackout for safety reasons, however battery systems can.
Solar panels are not designed to continue generating energy when there is a blackout. A standard grid-connected solar system is programmed to switch off to prevent the extra energy from being sent to the grid.
This is a safety mechanism to ensure no power from your panels can affect anyone working on the power lines. This means that even on the sunniest of days, a solar only system will not provide any energy to your home.
If you have a solar + battery system, you can almost say goodbye (and good riddance!) to blackouts. When there is a blackout, your system will operate as an isolated, off-grid system that can power a dedicated backup circuit in your home. Your solar panels will continue to generate energy to charge the battery, and your battery will provide energy to your backup circuit.
Note that in some models, you may need to engage your backup circuit if you have it bypassed.
If you have a battery only system, you also get the benefits of keeping the lights on when the grid is down. The difference here is that your solar panels will be switched off, so you'll be relying on the reaming charge in your battery to power your backup circuit.
In both cases, if you don't have charge in your battery then you won't have any power in a blackout.