A hard fork loosens or changes the rules so that blocks valid under the new rules can be invalid under the old ones. Nodes that do not upgrade reject those blocks, so unless everyone upgrades, the chain splits into two.
Examples
Bitcoin itself has avoided planned hard forks. Bitcoin Cash split off on 1 August 2017 by raising the block size limit, which created a separate coin. In March 2013, an accidental chain split happened when Bitcoin 0.8 accepted a block that 0.7 could not process. Miners reverted to the 0.7-compatible chain within hours.
Because every user would have to upgrade, hard forks are seen as risky and contentious in Bitcoin.