Hazard function ($h(t)$)

The hazard function is the instantaneous event rate conditional on survival to time $t$. For an absolutely continuous distribution, $h(t)=f(t)/S(t)$.

The cumulative hazard is $H(t)={\int }_0^{t}h(u)du=-\log S(t)$ for an absolutely continuous event-time distribution.

A hazard ratio compares two hazard functions at the same time. Under proportional hazards it is constant in time.