A common theme in the horror genre is of a monster that was just defeated coming back stronger than ever. Undying is the keyword mechanic equivalent of just that. A creature with undying will be returned to play with a +1/+1 counter on it after it dies, as long as it did not have a +1/+1 counter on it when it went to the graveyard. In most cases this is pretty straightforward, but there can be a few confusing situations that may arise.
Undying returns the creature to play under its owner’s control. If you gain control of an undying creature owned by an opponent, when it dies and returns to play it will not return to play under your control.
Undying only appears on creature cards, but the ability itself will work with any permanent type. If an undying creature stops being a creature, the undying ability will still return it to play. Similarly, if a permanent that is not normally a creature becomes one and gains undying, it will return with a counter even though it’s not a creature at that time.
Because token creatures cease to exist when they leave the battlefield, undying can never bring them back to play.
As mentioned above, the undying ability will only trigger if an undying creature leaves play without any +1/+1 counters on it. When a creature has both a +1/+1 and a -1/-1 counter on it, both will be removed when state-based actions are checked. But if an undying creature has a +1/+1 counter and receives any number of -1/-1 counter that lower its toughness to zero or cause the creature to now have lethal damage marked on it, then this creature will not be returned to play. This is because all state-based actions are checked at once. This means that this creature will both have the counter removed from it and go to the graveyard simultaneously. The undying ability uses last known information, and will look at this creature when it was in play. At that point it did have a +1/+1 counter on it, so it will not be returned to play. In other words, there is no point at which that creature was both on the battlefield and had no +1/+1 counters on it before it died.