Proportional hazards is an assumption of the Cox proportional hazards model of survival analysis and some other models as well. The assumption is that a linear increase in the predictor will have a uniform multiplicative relationship with the hazard.
In ordinary least squares regression, we assume that a linear change in an independent variable is related to a linear change in the dependent variable (note that the independent variable may be a transformed one). In proportional hazard models we assume that a linear increase in the independent variable is related to a multiplicative increase in the hazard. So, if the hazard when $x_1 = 2$ is double that when $x_1 = 1$ then we assume that the hazard when $x_1 = 3$ will be double that when it is 2.
One way of checking this assumption graphically is to look at the survival curves for multiple strata of the independent variable. If the assumption is true then the lines will be parallel.
