The hazard rate is the instantaneous rate at which units that have survived until time $t$ die at $t$. The hazard ratio is the ratio of 2 specified hazards.
Hazards are a central construct in survival (reliability) analysis. The hazard rate at time $t$ is the instantaneous rate of death (or failure, etc.) at $t$, conditional on survival until $t$. For a probability density function $f(t)$ and cumulative distribution function $F(t)$ the hazard rate function $\lambda(t)$ is given by:
$$
\lambda(t)=\frac{f(t)}{1-F(t)}
$$
The hazard ratio is the ratio of two specified hazard rates. A typical example would be the hazard rates for treated vs. untreated patients.
