A hypocentre can be various things depending on context. In the context of earthquakes, its meaning is not difficult to guess from etymology: Greek "hypo" means "below", so the hypocentre is the location where an earthquake originates, under the surface of the Earth's crust. The point on the surface directly above the hypocentre is, of course, the epicentre. The hypocentre is sometimes also known as the focus of the earthquake.
It's possible to calculate or estimate the location of a hypocentre by making measurements of the seismic waves it generates, but it's difficult to do so, since precision and accuracy is lost as the waves dissipate by the time they're measured.
In an entirely different context, a hypocentre can be the point directly underneath a nuclear detonation. Although any explosion in the air can have a hypocentre underneath it on the ground, the term only really became relevant with nuclear explosions. Hiroshima, for example, has a memorial at the hypocentre of the explosion. In this sense, it's essentially a technical synonym for the popularised phrase ground zero.