Hard Fork
In short: A hard fork is an irreversible blockchain upgrade that changes the protocol rules in a way older software does not accept.
A hard fork is an irreversible blockchain upgrade that changes the protocol rules in a way older software does not accept. Once it activates, every node has to upgrade to keep following the new chain.
Why it matters
Blocks made under the new rules look invalid to nodes still running old software, so a hard fork either brings the whole network along or splits it in two. When nearly everyone upgrades, as in a planned network upgrade, the chain simply continues with new features or fixes. When a meaningful part of the community rejects the change, both versions keep running as separate blockchains with separate coins. Exchanges usually pause deposits and withdrawals of the affected coin around the activation block.
Example
After an early DAO on Ethereum was drained through a smart contract exploit, the community adopted a hard fork that moved the affected funds to a contract from which the original depositors could reclaim them. Most users followed the upgraded chain, which kept the name Ethereum. A minority kept the original rules, and that chain continues as Ethereum Classic.