A hard fork is a change to the underlying software (Bitcoin/Altcoin) protocol with new rules that (might) improve the system. These new rules are usually modifications in data structures, different difficulty or an increase of valid transactions.
This protocol change means a permanent divergence is created in the blockchain.
The original or ‘parent’ blockchain and the forked one are identical up to the specific block number, where the hard fork occurred, but after that they diverge. You can think of it as follows:
A hard fork is a copy of its parent up to a specific block number, but after that, it differs.
Thus, if at the time of the hard fork, you ‘owned’ 10 coins on the parent blockchain, then you will have ten coins on the hard fork chain as well. And, probably, you can spend both without any problem.
Hard forks are nothing but rare in cryptocurrencies.