A Hydro-electricity power (HEP) plant is the place where the hydro electricity is produced and storaged. There are three types of it:   
- impourdment: is the most common type and it uses a dam to store river water in a reservoir.
- diversion: is called run-of-river. It facilities channels a portion of a river through a canal or penstock. It isn't necessary a dam.
- pumped storage: combines a small storage reservoir with a system for cycling water back into the reservoir after it has been released through the turbine, reusing the same water to generate electricity.

Here is an example from an Argentinean HEP and an international one:

※ In Argentina there are many hidroelectric dams and we investigated about one of the big ones called "Piedra del Águila " located between Rio Negro and Neuquen, at the Northwest of Comahue. It's 590 meters above sea level and it measures 172 meters high and 800 meters long. The water comes from the Limay and Collón Curá River. The dam is used for hidroelectricity generation and the regulation of river's flow. It provides energy to Argentina, who can sell it to the border countries. Also, a lot of people have to be relocated because of the dam's construction and it alterates aquatic fauna and water temperature.

※ The Guri power project, also known as the Simón Bolívar hydroelectric power station, ranks as the world’s third biggest one, with a capacity of 10,200MW. The Venezuelan power facility is located on the Caroni River in the Bolívar State of southeastern Venezuela. 

 The construction of the power project started in 1963. It was built in two phases, with the first phase completed in 1978 and the second phase in 1986. The power plant consists of 20 generating units of different capacities ranging between 130MW and 770MW. 

 Was awarded two contracts in 2007 and 2009 to refurbish four 400MW units and five 630MW respectively. The Guri power station supplies around 12,900GW/h of energy for Venezuela.

Finally here is a video explaining how a HEP plant works:

  https://youtu.be/OC8Lbyeyh-E

 Piedra del Águila Dam: https://images.app.goo.gl/XqcQUrTa5XDP9K4j8
Guri Dam:
https://images.app.goo.gl/NUbc9vQtHnneyZ4h8

The HEPs plants are formed by:

It is the largest plant built exclusively in Argentina on March 1985, the construction works of the civil works of this complex began, being the largest undertaking initiated by the state company HIDRONOR. The reservoir generates hydroelectricity and regulates the course of river.

 The Piedra del Águila Hydroelectric Complex is located on the Limay River.  Although it was originally designed to provide irrigation and generate electricity, it is now almost exclusively used as a hydroelectric generator.
https://www.patagonia.com.ar/Piedra+del+Aguila/893_Hidroeléctrica+Piedra+del+Águila+S.A..html
https://www.britannica.com/science/hydroelectric-power

Many dams were built for other purposes and hydropower was added later. In the United States, there are about 80,000 dams of which only 2,400 produce power. The other dams are for recreation, stock/farm ponds, flood control, water supply, and irrigation.

Hydropower plants range in size from small systems for a home or village to large projects producing electricity for utilities. The sizes of hydropower plants are described below.

The most common type of hydroelectric power plant is an impoundment facility. An impoundment facility, typically a large hydropower system, uses a dam to store river water in a reservoir. Water released from the reservoir flows through a turbine, spinning it, which in turn activates a generator to produce electricity. The water may be released either to meet changing electricity needs or to maintain a constant reservoir level.

A diversion, sometimes called run-of-river, facility channels a portion of a river through a canal or penstock. It may not require the use of a dam.

Another type of hydropower called pumped storage works like a battery, storing the electricity generated by other power sources like solar, wind, and nuclear for later use. It stores energy by pumping water uphill to a reservoir at higher elevation from a second reservoir at a lower elevation. When the demand for electricity is low, a pumped storage facility stores energy by pumping water from a lower reservoir to an upper reservoir. During periods of high electrical demand, the water is released back to the lower reservoir and turns a turbine, generating electricity. 

Facilities range in size from large power plants that supply many consumers with electricity to small and micro plants that individuals operate for their own energy needs or to sell power to utilities.

Although definitions vary, DOE defines large hydropower as facilities that have a capacity of more than 30 megawatts (MW).

Although definitions vary, DOE defines small hydropower as projects that generate 10 MW or less of power.

A micro hydropower plant has a capacity of up to 100 kilowatts. A small or micro-hydroelectric power system can produce enough electricity for a home, farm, ranch, or village.

Take a look at this diagram (courtesy of the Tennessee Valley Authority) of a hydroelectric power plant to see the details:

The theory is to build a dam on a large river that has a large drop in elevation (there are not many hydroelectric plants in Kansas or Florida). The dam stores lots of water behind it in the reservoir. Near the bottom of the dam wall there is the water intake. Gravity causes it to fall through the penstock inside the dam. At the end of the penstock there is a turbine propellor, which is turned by the moving water. The shaft from the turbine goes up into the generator, which produces the power. Power lines are connected to the generator that carry electricity to your home and mine. The water continues past the propellor through the tailrace into the river past the dam. By the way, it is not a good idea to be playing in the water right below a dam when water is released!
As to how this generator works, the Corps of Engineers explains it this way:
"A hydraulic turbine converts the energy of flowing water into mechanical energy. A hydroelectric generator converts this mechanical energy into electricity. The operation of a generator is based on the principles discovered by Faraday. He found that when a magnet is moved past a conductor, it causes electricity to flow. In a large generator, electromagnets are made by circulating direct current through loops of wire wound around stacks of magnetic steel laminations. These are called field poles, and are mounted on the perimeter of the rotor. The rotor is attached to the turbine shaft, and rotates at a fixed speed. When the rotor turns, it causes the field poles (the electromagnets) to move past the conductors mounted in the stator.

Pumped storage: Reusing water for peak electricity demand

Demand for electricity is not "flat" and constant. Demand goes up and down during the day, and overnight there is less need for electricity in homes, businesses, and other facilities. For example, here in Atlanta, Georgia at 5:00 PM on a hot August weekend day, you can bet there is a huge demand for electricity to run millions of air conditioners! But, 12 hours later at 5:00 AM .... not so much. Hydroelectric plants are more efficient at providing for peak power demands during short periods than are fossil-fuel and nuclear power plants, and one way of doing that is by using "pumped storage", which reuses the same water more than once.

Pumped storage is a method of keeping water in reserve for peak period power demands by pumping water that has already flowed through the turbines back up a storage pool above the power plant at a time when customer demand for energy is low, such as during the middle of the night. The water is then allowed to flow back through the turbine-generators at times when demand is high and a heavy load is placed on the system.
The reservoir acts much like a battery, storing power in the form of water when demands are low and producing maximum power during daily and seasonal peak periods. An advantage of pumped storage is that hydroelectric generating units are able to start up quickly and make rapid adjustments in output. They operate efficiently when used for one hour or several hours. Because pumped storage reservoirs are relatively small, construction costs are generally low compared with conventional hydropower facilities.