In the world of Bitcoin and Altcoins, a Hard Fork is defined as a drastic change to the protocol of a Blockchain that makes what would once been considered an invalid block or transaction valid. This creates something of a crossroad in the Blockchain (more on that here) — or rather, a proverbial fork in the road — in which one path follows the new, upgraded protocol and the other follows along the old path.
Let's spin that into something more digestible for us mere mortals. Take Bitcoin and Bitcoin SV, for example. The former came first in January 2009, setting the initial standard. Bitcoin SV then came along in November 2018, altering some core protocols, thus it was built on a Hard Fork of Bitcoin, because it's a permanent divergence from the previous version of the Blockchain that's used to drive Bitcoin — they do not intertwine.
To boil that down even further, a Hard Fork is no different to someone having an idea and creating a product around it, then someone else thinking of a way to put a spin on it (improving it, in some cases) and releasing their own version. The two operate independently, but the actual logic between them is similar, with the second iteration being built on the same core platform as the first, with some changes here and there.