Grid connected systems or on-grid systems, are solar systems that are connected to the building mains directly. These systems give priority to the solar power generated and only if the generated power is less than that required, will the grid power be used. As explained above, inverters are used to convert the produced DC electricity to AC. In case you opt for a grid connected system, you are eligible for Net Metering, a process where all the excess power that you produced is fed back to the grid and you receive a credit for the same. In case you generate more than you consume in one complete year, the DISCOM also pays you for the excess units generated. In many states, Net Metering has been made mandatory to avail subsidy for solar.
Off-grid solar systems are different from the on-grid system in the sense that they are set up independently from the state grid. These systems have an additional battery backup system which is charged by the generated solar energy. Batteries are charged when the generated power is more than the power being consumed. While on paper, battery backed systems sound like a great idea, these systems are generally more costly than on-grid systems and have lower efficiency. The battery needs to be replaced every five years or so but on the positive side, you can use solar energy even in the hours where there is no generation. Off-grid systems are perfect for your rooftop solar plants if you have a very erratic power supply or do not get any power at all.
A hybrid system brings the best of both worlds and is a combination of an on-grid and off-grid solar system. Hybrid systems have a battery backup which helps store the generated energy. What differentiates hybrid systems from off-grid systems is that, once the battery is fully charged, any excess power generated which is not being utilised will be fed back to the grid and the you will get suitably compensated. A hybrid system can also have other sources of energy producers, such as Diesel Generator (DG) or Wind coupled in the system.