Hard Fork
A rule change that makes previously invalid blocks valid, so every participant must upgrade or end up on a different chain.
Because non-upgraded nodes reject the new blocks, a hard fork requires coordination: clients ship the change, the activation is set at a block height or epoch, and operators update before that point. When everyone upgrades, the result is a single chain with new rules, which describes most of Ethereum's scheduled upgrades including the Merge. When a group refuses, two chains continue from the same history with separate coins and separate communities, as happened with Ethereum and Ethereum Classic after the 2016 DAO fork and with Bitcoin and Bitcoin Cash in 2017. A split also creates practical problems such as replay protection, since a transaction signed for one chain may be valid on the other.
In practice
After the 2016 DAO fork, both resulting chains shared every block up to the fork point, and balances existed on each of them separately.
The common misunderstanding
A contentious split is one possible outcome of a hard fork, not its definition; the great majority are routine coordinated upgrades that produce no new coin.