They operate by using high availability software to harness redundant computers in groups or clusters that provide continued service when system components fail.
HA clustering remedies this situation by detecting hardware/software faults, and immediately restarting the application on another system without requiring administrative intervention, a process known as failover.
[1] HA clusters are often used for critical databases, file sharing on a network, business applications, and customer services such as electronic commerce websites.
The attached diagram is a good overview of a classic HA cluster, with the caveat that it does not make any mention of quorum/witness functionality (see above).
HA clusters usually use all available techniques to make the individual systems and shared infrastructure as reliable as possible.