Occasionally a leap second is added or removed, to account for various sources of drifts. This leap second is taken to be the last second of the day where it is inserted, so the time technically either goes 23:59:59→23:59:60→00:00:00 or 23:59:58→00:00:00, depending on if a leap second is being added or removed.

That being said, many OSes handle this in such a way that 23:59:60 doesn't appear, either by repeating 23:59:59 (so it last two seconds), or "smearing" the extra second over the course of the day (so each second on the system clock that day is just over a second long).

Edit: I'm pretty sure I read recently that they are no longer going to be adding/removing leap seconds, although I'm not 100% sure that was a definitive decision.