Before I begin to answer this, it’s important to first give the definition of renewable energy.
Renewable energy is any form of energy that is sustainable and doesn’t “run out”. For example, energy from the sun is considered renewable. Energy produced from the motion of wind and water is considered renewable. Even geothermal energy available from intense heat found deep underground is considered renewable. These are forms of energy abundantly available in nature and as such are not something that can be “exhausted”. They are basically there for the taking 24/7. This is unlike fossil fuels (for example), which are a non-renewable energy source.
Fossil fuels are a form of “stored” energy created by the natural decay of plant and organic matter over millions of years. There were very unique geological conditions that took place in earth’s long history, which allowed the formation of fossil fuels. And as such, they are a one-time gift from nature. So when they run out they are gone for good! This is the main reason why it’s very important to make the switch to renewable sources of energy asap, while there are still enough fossil fuels in the world to drive the industrial engine that can put in place the necessary infrastructure needed for a renewable energy society. And once the infrastructure is in place it will be self-sustaining; meaning that the goods produced by renewable energy will be sufficiently integrated so that society no longer needs fossil fuels to function. The main advantage of renewable energy thus becomes clear: It can power our societies indefinitely!
Another advantage of renewable energy is that it is an inherently cleaner way to produce energy. There will no longer be any harmful emissions coming out of tailpipes and smoke stacks. Smog alerts will be a thing of the past. This points to the clear health benefit, which means a significant reduction in health care costs due to air pollution.
The world’s energy distribution will become much more balanced and democratic. This will help reduce monopolies on energy where the hands of a few control the main energy resource (currently the case with oil). This will mean more energy security for more countries. Countries which are now very reliant on foreign oil, can in some cases become completely energy independent due to their abundant renewable energy resources (such as wind and solar).
Furthermore, people (homeowners in particular), will have more control over their energy bills. Using home solar panels and wind turbines, as well as other potential renewables, they can reduce their reliance on the grid. In some cases, large centralized power generation will give way to more localized power generation, where small towns and cities set up their own local power generation using renewables. This is inherently a more democratic system anyway, and is a nice benefit of renewable energy. The nature of it means that many more people will be able to take matters into their own hands when it comes to how and where they get their power.
Gouging at the gas pumps will also be a thing of the past. If the energy comes from renewables there will be no such thing as “mysterious” market forces driving the price of “gas” up. In this case the “gas” will be the electricity used to charge your (say) electric car.
Lastly, renewables will make people much more conscious of how much energy they consume. Given the nature of renewables, they can’t simply be placed in one central location where they will be out of sight and out of mind (like a coal or nuclear plant). They will be placed where the sun shines, the wind blows, and where people will accept this as a natural part of taking responsibility for their energy consumption.
To get a detailed description on the type of renewable energy sources available see Renewable Energy Sources Primer.