Quantum mechanics is a branch of science that started in the early 20th century, as scientists made certain observations that did not align with classical physics and chemistry's prediction on the behaviour of atoms (and their components). Sometimes, it's known as the '30 years that shook physics', this revolutionary theory led to the idea that energy must be quantized. That means that energy exists in discrete packets of predetermined value, rather than continuous value. A proper treatment of quantum mechanics is not only beyond this encyclopedia, but takes many years of study.
Quantum mechanics plays into how photovoltaic cells work for solar power and how nuclei behave for nuclear power. The video below gives a historical introduction to quantum mechanics from the brilliant Physics Girl.
For more on quantum mechanics, a good place to start is hyperphysics for a physics focus. For a chemistry focus please see UC Davis's chemistry wiki. For related concepts within energy science please try: