"A hard fork is generated when the cryptocurrency’s developers decide that changes must be made to the programming of the coin that will create incompatibilities between the older and newer version. When the changes are made, all users of that coin must willing to update all applications to continue to use that coin type correctly. "
A hardfork is basicaly a radical change of the current protocol. With a hardfork, all nodes or users needs to upgrade the latest versions of the protocol because nodes running previous versions will no longer be accepted by the newest version.
Why we need to upgrade the current protocol?
1- To fix important security risks found in older versions.
2- To add new functionality.
3- To reverse transactions.
HARDFORK CASES
1- Bitcoin Cash (BCH)
It was a hardfork of Bitcoin. The difference was higher transaction speed with a less decentralized way.
2- Bitcoin Classic
In 2016, a group of developers launched Bitcoin Classic. It was a hardfork of Bitcoin. Block sizes increased to 2 megabytes.
3- MoneroV (XMV)
It was a hardfork ofvMonero which has infinite coin supply. MoneroV caps the total supply at 256 Million XMV tokens that shall be reached by 2067.
4- Ethereum Classic (ETC)
"Ethereum Classic is the first hard fork of the original Ethereum blockchain, and it was a contentious hard fork. The hard fork was a result of a difference in ideologies. The split occurred in 2016 after a decentralized pooling fund – called a Decentralized Autonomous Organization (DAO) – that as built on Ethereum was hacked."
Sources:
https://www.tech-recipes.com/rx/48517/cryptocurrency-what-is-a-fork/
https://cointelegraph.com/bitcoin-cash-for-beginners/what-is-hard-fork#terms-you-should-know
https://www.investopedia.com/terms/h/hard-fork.asp
https://www.coindesk.com/information/hard-fork-vs-soft-fork/
https://www.fxcm.com/insights/what-is-a-hard-fork/
https://www.investopedia.com/tech/history-bitcoin-hard-forks/
https://masterthecrypto.com/ethereum-hard-forks-guide-ethereum-classic-etherzero-metropolis/