Off-Grid/Hybrid Solar Definition: These systems allow you to store your solar power in batteries for use when the power grid goes down or if you are not on the grid. Hybrid systems provide power to offset the grid power whenever the sun is shining and will even send excess power to the grid for credit for later use. Benefits: Provides power for your critical loads when the power grid is down. Downside: Cannot be expected to provide power for all your loads since the cost and volume of batteries would be prohibitive. Off-Grid systems require a lot more specialized equipment to function that is more costly and more complex to install. Specifically they require a central/string inverter, a charge controller as well as a batteries.
Sizing the solar array and the batteries required is complex. Detailed analysis of your requirements will be needed to provide for your minimal critical needs. You'll also need to rewire you main electrical panel to isolate the "critical loads" so that only they are provided power in an outage. This means that your well pump, refrigerator and a few lights are provided power while your air conditioners and TV and other non-essential loads are not.
This is definately more complex to install as well. There are dangerious components, mostly dealing with high amp batteries so caution needs to be exercised. Also, batteries are expensive, require ongoing maintenance and periodic replacement.