Introduction: Stratified Cox models are like ordinary proportional hazards Cox models, but the stratifying variable can have different baseline hazards across categories.
Typical application: In medicine, such models are often applied as extension of Cox models where for some variables proportional hazards do not seem to hold. These can then be used as strata.
Interpretation of coefficients: Say we have fitted a stratified Cox model to estimate the effect of sex on infection, where we stratified by hospital. We get a hazard ratio of two for male. This then means that for sex =male, the risk is 2 times higher to get an infection.
Main assumption: The proportional hazards assumption still applies to the variables whose coefficients are estimated.