Organic molecules are molecules which are made of carbon and other elements. At least one of the atoms in the molecule must be carbon, they usually involve hydrogen and oxygen, and they can also involve nitrogen, sulfur, and phosphorous. Hydrocarbons, like alkanes, alkenes and alkynes are all organic molecules and so are alcohols, carboxylic acids and carbohydrates.
Most (85% or so) of the world's primary energy comes from fossil fuels, which are made up of mostly organic molecules. These organic molecules undergo combustion - they react with oxygen from the atmosphere and produce carbon dioxide. This process typically involves an exothermic reaction which releases heat energy that is converted into usable energy, usually with a heat engine.
Organic chemistry is a vast field that takes many years of study to fully understand; to get a start, please see UC Davis's chem wiki.