The survival function answers “what is the probability this thing lasts beyond time t?” for a random lifetime: a customer staying subscribed, a patient staying alive, a component staying functional. Plotted against time it is the survival curve, and reading values off that curve is how expected lifetimes, medians, and lifetime values get computed.
It is tightly linked to the hazard function: survival to time t is the product of one-period conditional survival probabilities, one minus the hazard at each step. Estimating the survival function from data where many lifetimes are still unfinished is the job of estimators like Kaplan-Meier.