What is the science behind hydroelectric energy?

The science behind hydroelectric power includes elements of physics. Given an existing dam with a reservoir and a power station, let's follow the action. Water is allowed to build up behind a dam. This creates potential energy. Releasing the water and directing it through turbines converts the potential energy into kinetic energy. This energy, which is mechanical energy, spins the turbines. Call that the "first stage" of the process.

The turbines are coupled to generators, and the mechanical energy of the turbines is converted into electromagnetic energy in those generators. The electromagnetic energy (or just "electricity" if you prefer) is routed to a transformer bank to step it up in voltage. The higher voltages allow for less power loss during transit, and the energy is transported to points of use. There, it is stepped down to relatively "safe" voltages and routed out to electrical equipment and devices.