Intuitively, a stopping time is simply a random variable that determines when to stop a random process. Naturally, the decision to stop the process should be based on only information until that time, not any future information.
Mathematically, a stopping time is a random variable on a probability space with filtration such that for all , the event is is measurable. Eg suppose you have a stochastic process and . Then the decision to stop the process at time should be based only on , not on for any .