Hydropower
As rivers flow down from the mountains, the water is constantly losing potential energy
(that is exactly why water flows down hill). We can capture some of that energy. This is basically solar power, because it was the sun that evaporated the water from the oceans that later rained out over the mountains. It's renewable energy – we expect the river to continue to run for a long time. Once the dam has been built, it is a source of cheap power.
This diagram shows a dam in cross-section:
Water that is trapped behind the dam flows through the turbine, which drives the generator and produces usable energy, usually in the form of electrical power.

Here is another diagram:
The two diagrams seem to differ in that in the first one the "penstock" that carries the water to the turbine is deep below the surface of the lake behind the dam, while in the second diagram it is closer to the surface. This has no effect on how the dam works: as water goes through the turbine, the level of the lake goes down, so that the effect always is to move water from the surface of the lake down to the level of the stream below the dam. The height difference between the two levels determines the amount of potential energy that is being released.
Potential energy =
9.81 Joules/kg-m *mass*height
Since 1 cubic meter of water is 1000 kilograms, we can also write
Potential energy =
9810 Joules/m^4 * volume * height
The amount of energy stored by a dam is determined by the height of the dam and the volume of water behind it.

For example, consider Hoover Dam
The height difference is 221 meters
(almost 700 feet), and there are 25 cubic kilometers of water behind it.
When 1 cubic meter goes through the turbine, the amount of energy released is
Energy = 9810 J/m^4 * 1 m^3 * 221 m
= 2,100,000 Joules
= 0.6 kWh
(remembering that a kWh = 3,600,000
Joules)
= 6 cents worth of electricity
(because a kWh costs about 10 cents).
Since
25 km^3 = 25 * (1000 m)^3
= 25 * 10^9 m^3 , the lake behind Hoover Dam represents 5 * 10^16 Joules – 20 billion dollars worth!
(25*10^9 * 2,100,000 = 5 * 10^16)
The amount of power coming from the dam is the amount of energy released per second:
Power = 9810 J/m^4 * height * (volume/time)
It depends on the flow rate – the amount of water per second. The generators at Hoover dam can produce 2 gigawatt when running at maximum speed; since each cubic meter gives 2 kiloJoules, the flow rate must be 1000 cubic meters per second. It's a river! On the average, this water is replaced by the Colorado River, which flows into the lake above the dam.
Disadvantages of hydropower
So why don't we get all of our power from dams? Because there aren't enough rivers!
We build dams for several reasons:
*Hydropower. There are dams on the Columbia River in Washington state that provide a lot of the power for that region; some countries (like Venezuela and Norway) get almost all of their electrical power from hydropower.

*Flood control. Rain and snow don't come at a regular rate. Snow lies on the ground and then all melts in the spring. This can cause seasonal flooding. The dams on the
Tennessee River were built in part to control the river flow.
*Water conservation. In between the wet seasons are dry seasons. Cities, agriculture, and industry will need water during the dry season; storing water behind the dam provides for this. However, the people below the dam may complain if the dam conserves the water so well that they don't get any. This is particularly a problem when a river flows from one country into another. For example, the Colorado River that is supplying Hoover Dam goes into Mexico and never makes it to the ocean at all: all the water gets used.
*Make rivers navigable. There are small dams along the Kentucky River – every 20 miles, near Lexington – which were originally intended to make the river deep enough that boat traffic could move along the river in all seasons. At each dam there was a
"lock" – a kind of elevator to move boats past the dam.
*Water recreation value. Lakes can be used for boating, fishing, and are pretty. People build houses by lakes. The lakes of Kentucky were all made by damming a river, and the local economy has benefited from this.
Let us note that these uses are not entirely compatible. The water level behind a dam built for flood control or water conservation will go up and down with the seasons; hydropower will take water from the dam at the time power is needed, which may not be when water is flowing into the lake. Meanwhile, the boaters and landowners would like the water level to stay pretty much the same, all the time.
There are other problems that limit how much hydropower can be used.
*A dam floods a valley, which is the best place for agriculture; usually there are already people living in the valley, and they are not happy about having to move.
* The dam will silt up. Rivers carry mud downstream. The water in the lake behind the dam is not flowing very fast, and the mud settles out. In a few hundred years, the lake behind Hoover Dam will completely fill up: 221 meters of dirt! But the power generated depends only on the height difference between the lake and the bottom of the dam, so full power production will continue until almost the end.
* What if the dam breaks? All that energy could become a powerful flood. This has happened – not too often, and mostly involving small dams. Where there are cities downstream from a large dam it is hard to imagine what would happen.
* What if there is no water in the river? Venezuela has this problem right now. There has been a drought for quite a while, and the dams can no longer function. Since
Venezuela was getting most of its electrical power from the dams, this is a big problem.
Despite all of the objections, hydropower is a good idea. Dams have large initial cost, but produces power for a long time afterwards
Dix Dam created Lake Herrington. It was originally built for hydroelectric power, and occasionally still does. But it is only 24 Megawatts, which is tiny compared to the 750
Megawatt coal-fired power plant right next to it.

There is another hydropower installation only a few miles away from Dix Dam on the
Kentucky River at Lock #7. This is a little dam that was built to make the river navigable; it's less than 10 meters (30 feet) tall. It generates 2 Megawatts, which is only
1/1000 of Hoover Dam, but it surely cost 1/1000 less to build. Here is a picture of the power station, which is just below the dam
There is a man sitting in a chair on top of one of the three pillars, right next to the windows. The power plant is raised above the river, in case there is a really big flood.
Part of the dam is visible to the left.

Here is an aerial view of the lock and dam. The river is flowing upwards; the white band is the water that has flowed over the dam. At right you can see the lock: it is a big tub with doors (but one of the doors is missing, and the lock no longer works). A boat could get past the dam by moving into the tub, they would close the doors, and then let water in or out to raise or lower the boat. The power plant is the dark square at the left of the dam.