There is not one equation or one method to do this. For a comprehensive overview of survival analysis methods, definitions, and equations see Hosmer, Lemeshow, and May's book "Applied Survival Analysis". All the definitions below are described and given with equations.
For parametric survival regression (like exponential or Weibull), this is trivial. The hazard is explicitly modeled and can be summed up to provide estimates of survival.
For semi-parametric survival regression using the Cox model, the hazard is not modeled explicitly. The model coefficients are either used to obtain a continuous linear predictor (which can rank observations in terms of a risk-profile but does not predict risk), or are rounded off to create a convenient point-based scoring profile. The baseline cumulative hazard function is estimated using a Breslow-step estimator, and an absolute risk over a certain time period is given by summing up the product of the cumulative hazard and the exponentiated linear predictor.
A good example is described here for the Framingham stroke risk score.