Ethereum (ETH) and Ethereum Classic (ETC) is a good example of a hard fork.
The Decentralized Autonomous Organization planned to build a crowdfunding platform based on Ethereum ecosystem. It was officially released in May 2016 and ended in June of that year, raising more than $160 million. But afterward, the DAO was exploited by hackers and lost Ethereum worth $50 million. The Ethereum community voted to alter the Ethereum code, in the hope that the investor’s funds can be recovered. Therefore, Ethereum hard forked at block 199000, rolling back the Ethereum transactions (including the hacker’s). However, some people believe that this violated the decentralized and immutable feature of the blockchain, and insisted on mining on the original chain. This resulted in two different chains being formed. One of the chains that does not recognize the rolled back transactions is ETC and the other is ETH. Each of them represents a different community consensus and values.