In survival analysis, the hazard rate at time $t$ is the instantaneous probability of death at $t$, conditional on survival until $t$.
In survival (reliability) analysis, the hazard rate at time $t$ is the instantaneous probability of death (or failure, &c.) at $t$, conditional on survival until $t$. For a probabilty density function $f(t)$ and cumulative distribution function $F(t)$ the hazard rate function $\lambda(t)$ is given by $$\lambda(t)=\frac{f(t)}{1-F(t)}$$