As it relates to blockchain technology, a hard fork (or sometimes hardfork) is a radical change to the protocol that makes previously invalid blocks/transactions valid (or vice-versa), and as such requires all nodes or users to upgrade to the latest version of the protocol software. Put differently, a hard fork is a permanent divergence from the previous version of the blockchain, and nodes running previous versions will no longer be accepted by the newest version. This essentially creates a fork in the blockchain, one path which follows the new, upgraded blockchain, and one path which continues along the old path. Generally, after a short period of time, those on the old chain will realize that their version of the blockchain is outdated or irrelevant and quickly upgrade to the latest version. Hard forks refers to the underlying blockchain of a digital currency forks into 2 chains initially due to the underlying rules change. After a hard fork, the blockchain conforms to the revised (upgraded) rules is only considered and accepted by nodes that support the new rules, whereas the original blockchain conforms to the original rules is only recognized by nodes that support the original rules. That is, the new rules are not compatible with the old existing rules. Think of hard fork as a possible permanent divergence of a block chain that might result in splitting a chain into two, thus producing an extra new coin for every existing coin.
DEFINITION of 'Hard Fork'
A hard fork can be implemented to correct important security risks found in older versions of the software, to add new functionality, or to reverse transactions (as in the case with the hard fork to reverse the hack on the DAO (decentralized autonomous organization) in the Ethereum blockchain).
BREAKING DOWN 'Hard Fork'
A hard fork involves splitting the path of a blockchain by invalidating transactions confirmed by nodes that have not been upgraded to the new version of the protocol software. Following the hack on the DAO, the Ethereum community almost unanimously voted in favor of a hard fork in order to roll back transactions that siphoned off tens of millions of dollars worth of digital currency by an anonymous hacker. The hard fork also allowed DAO token holders to get their ether funds returned to them.
There were many hard forks in Bitcoin history. Some of them failed for a reason or two - lack of consequent support, buggy software, When a hard fork failed to go on, the new chain wouldn't last too long after the split. For example, Bitcoin Unlimited, Bitcoin Classic, Bitcoin XT  are among the dead list.
Here is a list of successful hard forks events and the attributes of the new coin each fork introduced.
The proposal did not exactly unwind the network’s transaction history. Rather, it relocated the funds tied to The DAO to a newly created smart contract with the single purpose of letting the original owners withdraw them. DAO token holders will be able to withdraw ETH at a rate of approximately 1 ETH to 100 DAO. The extra balance and any ether that remains as a result of the re-entrancy exploit and the splitting mechanism will be withdrawn and distributed by the DAO curators, or individuals selected prior to the collapse of the DAO to provide "failsafe protection" for the organization.
Digital currency build upon protocol and rules.
Rule, what rule?
For example, original Bitcoin protocol/rule is capable of processing 3-7 transactions per second, which is way too low for Bitcoin to become a transaction currency. Therefore, they were many thoughts to solve this 'problem'.
Bitcoin has other 'problems' too. For another example, wouldn't it be better if miners can mine coins using GPU and not necessary need to have ASIC based mining hardware? In order to 'solve' this particular 'problem', new rules were proposed to change the mining algorithm to not in favor of miners using ASICs.
To date, proposals were put in action to solve these 'problems' involve increasing block size limitation (e.g. BCH, B2X), or segregating some non-transactoin data from the block (e.g. Segregated Witness). Hard fork can be implemented to correct important security risks found in older versions of the software, to add new functionality, or to reverse transactions (as in the case with the hard fork to reverse the hack on the DAO (decentralized autonomous organization) in the Ethereum blockchain).
HARD FORK THAT SUCCEEDED