Even after the nuclear reactor is shut down, the fuel continues to generate decay heat.
Decay heat results from the radioactive decay of fission products. The rate at which the decay heat is generated decreases the longer the reactor has been shut down.
So the longer the spent fuel has been out of the reactor, the less heat that it gives off.
Storing the spent fuel in a pool of water is a way to provide an adequate heat sink for the removal of heat from the irradiated fuel. In addition, the fuel is located far enough under water that the radiation emanating from the fuel is shielded by the water to adequately protect the workers from the radiation.

Yes. The regulations give criteria that must be met for fuel storage and handling. This includes designing fuel-storage systems to ensure adequate safety under normal and postulated accident conditions. The system is to be designed with suitable shielding for radiation protection, with appropriate containment, confinement, and filtering systems, and with a heat-removal capability that is reliable and that can be tested to ensure that it meets the requirements for removing the heat produced by the spent fuel.

All nuclear power plants have a reinforced-concrete spent fuel pool (SFP) structure designed to retain its function, even following the design-basis seismic event (that is, seismic Category 1 or Class 1 [earthquake]) that is anticipated for the area. The SFP also has a welded, corrosion-resistant liner.
All plants except for one have leak-detection channels positioned behind liner plate welds to collect any leakage and to direct the leakage to a point at which it can easily be monitored.
Nearly all nuclear power reactors have passive features preventing draining or siphoning of the SFP to a coolant level below the top of stored, irradiated fuel. Excluding paths used for irradiated fuel transfer, passive features at nearly all nuclear reactors prevent draining or siphoning the coolant to a level that provides inadequate shielding for fuel seated in the storage racks.

In the event that SFP coolant inventory decreases significantly, several indicators are available to alert operators to that condition. The primary indication is a low-level alarm.
A secondary indication of a loss of coolant is provided by area radiation alarms.
These primary and secondary alarms indicate a loss of shielding that occurs when SFP coolant inventory is lost. Except for the SFP located inside the containment building, the area radiation alarms are set to alarm at a level low enough to detect a loss of coolant inventory early enough to allow for recovery before radiation levels could make such a recovery difficult.

The level of radioactivity in the water of the spent fuel pool is low. In addition, nuclear plants have cleaning systems to maintain the purity of the water in the spent fuel pools. All nuclear plants have a groundwater monitoring system around the facility so that if a system leaks, there is a method for alerting the licensee to the problem as well as for providing information regarding the location of the contamination.

The consequences of losing the heat-removal capability or water (coolant) in a spent fuel pool depends on the amount of time since the fuel was last used for power operation inside the reactor.
If fuel was recently used for power operation, there may be enough decay heat to cause the spent fuel pool coolant to heat up to the boiling point if forced cooling were lost to the spent fuel pool.
If plant operators took no action, boiling would cause the level in the spent fuel pool to decrease over time. However, operators have redundant sources of water to add to the pool to maintain coolant level should a loss of forced cooling occur.
Operators are alerted to a loss of level condition by a series of alarms at the cooling system control station and in the main control room. Given the unlikely event that no operator action is taken, the pool level would decrease at a very slow rate (about one foot every several hours to weeks, depending on the age of the stored fuel).
The longer the time interval since the last batch of fuel was used to generate power in the reactor, the longer it would take for the spent fuel to boil off the spent fuel pool coolant.
Boiling the spent fuel pool coolant is, however, an acceptable method for cooling spent fuel and has a minimal effect on public health and safety. In the unlikely event that a large loss of coolant uncovers the spent fuel, if sufficient time has not elapsed since the fuel was used to generate power in the reactor, the spent fuel may have enough decay heat to overheat the cladding in air and cause it to ignite.
The resulting fire could carry radioactive particles offsite and the consequences could be significant.
However, the NRC staff considers this a very low probability accident because of design features required at all spent fuel storage pools that minimize the possibility of losing all of the spent fuel pool coolant.

Spent fuel pools are designed to withstand earthquakes greater than any earthquake that actually occurred or is expected to occur in the area of the plant. Therefore, the probability of the spent fuel pool rupturing due to an earthquake is very low. However, in the unlikely event that a very large earthquake does occur, one that is larger than the pool was designed to withstand, the pool structure could fail and allow the coolant to drain out. The consequences of an accident like this are discussed in the response to Question 5.8.4, above. In the unlikely event that an aircraft crashed into the spent fuel pool, the pool structure could be severely damaged and not capable of maintaining coolant level. In this event, consequences such as those discussed in Question 5.8.4 could result. However, the staff has evaluated the possibility of an aircraft impacting the spent fuel pool and consider it a very low probability event.

A cooling system removes decay heat from the spent fuel pool. The coolant in the spent fuel pool is maintained below a specific temperature and the level of the water is maintained at a specific height over the spent fuel. Temperature indicators are installed and are either equipped with an alarm or require visual surveillance on a daily basis. High/low-water-level monitors are installed in spent fuel pools. The monitor alarms at the spent fuel pool and in the control room when the spent fuel pool's water level is not within the specified limit.