Periodic
One example of something that is periodic in discrete time is the phase of the moon. This link describes the phase of the moon and how it changes. This is periodic because we can look into the future and predict which phase the moon will be in on any specific date. It's discrete time because we can't see the phase of the moon during the day.
Non-Periodic
An example of something that is not periodic in discrete time is the mile times of a marathon runner. This is not periodic because we can't look into the future and predict exactly how long it will take a runner to run their 13th mile before the race has even started. It is discrete time because we take our measurements in integers of seconds.
