Hard fork
Also called: hardfork, hard-fork, chain split, backward-incompatible upgrade.
A hard fork is a change to Bitcoin's consensus rules that makes previously invalid blocks valid, so nodes that don't upgrade split off the chain.
A hard fork loosens Bitcoin's consensus rules: blocks or transactions that old software rejects become valid. Nodes that don't upgrade reject the first block that uses the new freedom and keep following a chain without it, so a hard fork without near-universal adoption splits the network in two. BIP 99 defines it as a consensus fork "that makes previously invalid blocks valid" and notes that hard forks "require all users to upgrade".
Hard forks and the block size debate
Raising the 1,000,000-byte block size limit was the most discussed hard fork. Several BIPs described how it could be done: BIP 101 (8 MB, doubling every two years), BIP 102 and BIP 109 (one-time increases to 2 MB), and growth schedules or dynamic limits in BIP 100, BIP 103, BIP 104, BIP 105, BIP 106 and BIP 107. Bitcoin adopted none of them. Capacity was instead raised by SegWit's block weight rule, a soft fork. Projects that changed the block size by hard fork, such as Bitcoin Cash in 2017, became separate networks with their own coins.
Has Bitcoin ever had a hard fork?
Bitcoin has never activated a planned hard fork. The closest event is described in BIP 50, Gavin Andresen's post-mortem of March 2013. A block that Bitcoin 0.8 accepted was rejected by older versions because of an undocumented database lock limit, and the chain split. Mining pools downgraded to restore a single chain, and version 0.8.1 temporarily limited blocks so older nodes could keep up. BIP 50 records that on 16 August 2013 block 252,451 was accepted by the main network, forking unpatched nodes off it.
Every consensus change Bitcoin has deployed since has been a soft fork; the soft forks guide explains why.